During Neera Tanden’s confirmation hearings, senators questioned her social media posts and her work at a liberal think tank.In a statement, Mr. Biden said that Ms. Tanden had requested that her nomination for director of the Office of Management and Budget be withdrawn and that, while he agreed to do so, he planned to find a place in his administration for her to serve in a different capacity.

“I have the utmost respect for her record of accomplishment, her experience and her counsel,” he wrote, bowing to the reality of the first significant defeat of his presidency. “I look forward to having her serve in a role in my administration. She will bring valuable perspective and insight to our work.”

Ms. Tanden, who was a senior adviser to Hillary Clinton’s 2016 presidential campaign, had drawn bipartisan criticism for a prolific stream of social media posts that criticized lawmakers in both parties, often in vitriolic terms, and for her work at a liberal think tank, the Center for American Progress. Mr. Biden selected her to direct the budget office before Democrats had won control of the Senate, surprising lawmakers and aides in both parties.

The pick also surprised many of the economic aides in Mr. Biden’s inner circle, which Ms. Tanden had not been a part of, who saw her as more publicly combative and less bipartisan than most of Mr. Biden’s other nominees.

In a letter released on Tuesday by the White House, Ms. Tanden asked Mr. Biden to end her nomination, acknowledging the political opposition to her serving as the administration’s budget chief. “Unfortunately, it now seems clear that there is no path forward to gain confirmation, and I do not want continued consideration of my nomination to be a distraction from your other priorities,” she wrote.

A senior administration official said Tuesday night that Mr. Biden and Ms. Tanden had agreed to give up on the nomination after Senator Lisa Murkowski, Republican of Alaska, made clear to the White House earlier in the day that she would not vote for her.

The official said it was a “mutual understanding” that without Ms. Murkowski’s support, and with the public opposition to Ms. Tanden’s confirmation from Senator Joe Manchin III, Democrat of West Virginia, there was no longer a path forward in the Senate, which is divided 50 to 50 between Republicans and Democrats.

But asked on Tuesday evening on Capitol Hill if she had told anyone in the White House that she would vote against the nomination, Ms. Murkowski said she had not. “No, I never did,” she said. “They never asked.”

While Ms. Tanden’s social media posts were often aggressive, White House officials believed Republican senators would not scuttle a nomination over Twitter behavior after years of standing behind President Donald J. Trump, who excoriated Republicans and Democrats alike in frequent Twitter rants. The officials also thought senators would be drawn to the groundbreaking nature of Ms. Tanden’s nomination — she would have been the first Indian-American to lead the budget office and had a personal story of being raised by a single mother who at times relied on government assistance to get by.

Ms. Tanden and a variety of groups supporting her were able to secure several high-profile endorsements, including one from the U.S. Chamber of Commerce.

But the White House underestimated what would become bipartisan consternation over Ms. Tanden’s posts, particularly given Mr. Biden’s repeated calls for “unity” after four years of divisive rhetoric from Mr. Trump.

During two confirmation hearings, senators in both parties grilled Ms. Tanden about her social media posts and her decision to delete more than 1,000 tweets after the election in November. Among those who questioned her social media comments was Senator Bernie Sanders, independent of Vermont and the chairman of the Budget Committee, who singled out Ms. Tanden’s “vicious attacks” on him and the staff that supported his 2016 presidential campaign.

She apologized, but it was clear that some senators were not inclined to accept it.

That included Mr. Manchin, whose decision last month to oppose her nomination dealt a crucial blow to her chances of confirmation.

“I believe her overtly partisan statements will have a toxic and detrimental impact on the important working relationship between members of Congress and the next director of the Office of Management and Budget,” Mr. Manchin said at the time. “For this reason, I cannot support her nomination.”

Less than a week ago, top White House aides had vowed not to give up on Ms. Tanden, accusing Republicans of being overly sensitive about her online criticism of them and promising that the president intended to fight for his right to select his own advisers.

Ron Klain, the White House chief of staff and a champion of Ms. Tanden’s nomination to the budget post, said last week that the administration was “fighting our guts out” to get her confirmed. And Jen Psaki, the press secretary, said Mr. Biden was continuing to work the phones with senators to build support for her.

“The president nominated Neera Tanden because she is qualified, because she is experienced, because she has a record of working with people who agree and disagree with her,” Ms. Psaki said at the time.

One administration official said Tuesday that the aggressive push on Ms. Tanden’s behalf, including Mr. Klain’s television appearances, was in part intended to blunt criticism from the left wing of the president’s party for not fighting on behalf of a qualified woman of color who was mainly being criticized for her Twitter account.

Ms. Tanden apologized for her posts during her two contentious confirmation hearings, saying, “I’m sorry, and I’m sorry for any hurt they’ve caused.”

Her chances at winning the support of a majority of senators diminished rapidly in the weeks after those hearings, as centrist Democrats and Republicans — including Senators Mitt Romney of Utah and Susan Collins of Maine — announced they would not back her. That imperiled Ms. Tanden’s margin for confirmation in the 50-to-50 Senate, leading two committees to abruptly postpone votes last week on advancing her nomination.

With at least one Republican needed to join all of the Democrats, Republican leaders privately counseled their senators to remain united in opposition to Ms. Tanden. Ms. Murkowski, a moderate Republican who was thought to be a potential supporter, had not publicly announced her position, but it was not clear that there would be enough Democrats behind Ms. Tanden to confirm her even if she had won Ms. Murkowski’s backing.

Mr. Sanders was among the lawmakers who repeatedly declined to say how he would vote on Ms. Tanden’s confirmation, saying on CNN shortly before the White House pulled the nomination that “I will make that decision when the vote takes place.”

Ms. Tanden’s supporters continued to believe as recently as Tuesday that she would garner the support of every Senate Democrat besides Mr. Manchin if she were able to win Ms. Murkowski’s vote.

In an effort to secure that support, Ms. Tanden met with Ms. Murkowski on Monday. The senator walked Ms. Tanden through what she called her “Alaska Tutorial 101” — complete with maps of tribal lands, state waters and roadless areas — and outlined a series of concerns about the effect of some of the Biden administration’s economic proposals on her state.

Ms. Murkowski, speaking after the nomination was withdrawn, said, “I guess the message that it sends is that you really have to work your agenda, extra hard,” given the 50-to-50 split. “I think they probably thought well, OK, well we’ll have Manchin right? So we don’t need a Republican. Well, maybe it’s a lesson that you’re not always going to have Manchin.”

White House officials had remained adamant in their support for Ms. Tanden, who would have become the first woman of color to lead the Office of Management and Budget. Democrats also argued that Ms. Tanden was facing unfair scrutiny, particularly given that their Republican counterparts had spent years expressing no concern about the often offensive and racist posts made by Mr. Trump.

Ms. Psaki has declined to address questions about fallback nominees, including Gene Sperling, a former National Economic Council director, and Ann O’Leary, the former chief of staff to Gov. Gavin Newsom of California.

But some House Democrats have pushed hard in recent days for the White House to scrap Ms. Tanden’s nomination and substitute Shalanda Young, the first Black woman to serve as the staff director for Democrats on the House Appropriations Committee. She is Mr. Biden’s pick to be the No. 2 at the budget agency.

Some administration officials expect Mr. Biden to nominate Ms. Young to head the agency, though Mr. Sperling remains a contender. At least one top Republican, Senator Richard C. Shelby of Alabama, who worked with Ms. Young as chairman of the Senate Appropriations Committee, issued a statement this month declaring that she “would have my support, and I suspect many of my Republican colleagues would support her as well.”

At Ms. Young’s confirmation hearing before the Senate Budget Committee on Tuesday, lawmakers in both parties lavished her with praise for her work on Capitol Hill, helping negotiate legislation that ended the nation’s longest government shutdown in 2019 and a series of pandemic relief bills in 2020.

“Everybody that deals with you on our side has nothing but good things to say,” said Senator Lindsey Graham of South Carolina, the top Republican on the budget panel. “You might talk me out of voting for you, but I doubt it.”

“You’ll get my support, maybe for both jobs,” he noted.

As a candidate, Mr. Biden had argued that his longstanding relationships in Congress, where he served for 36 years as a senator, would allow him to corral bipartisan support for his policies and his personnel by appealing to Republicans and moderate Democrats as well as liberal stalwarts in his party.

But since taking office, the president has struggled to turn that promise into reality. His administration is pressing forward with a $1.9 trillion Covid relief package that appears unlikely to win much Republican support in the Senate. His efforts to pass an increase in the minimum wage to $15 per hour quickly faltered.

“I can’t imagine what it’s like being in the administration and trying to figure out, OK, how do we make something happen here,” Ms. Murkowski said. “They’ve got a pretty, pretty big learning curve right now.”