President Trump tried to channel the anger he has generated about his election loss into voter turnout for two Republican senators facing runoff elections next month, telling supporters at a Saturday rally that electing the GOP candidates would amount to “revenge” on Democrats.

With control of the Senate at stake in the Jan. 5 runoffs, Republicans were banking that Mr. Trump’s appearance here would drive up enthusiasm for the two GOP senators after his narrow loss in the state’s presidential election. But Mr. Trump also focused much of his speech on his unsupported claims of fraud in the November election, which some party officials say are raising distrust of the election system and could discourage voting.

“You know we won Georgia, just so you understand,” Mr. Trump said as he opened his remarks. “We didn’t lose.”

The initial count and a recount in Georgia found that Democrat Joe Biden won, and the Georgia secretary of state is expected to certify on Monday that a second recount confirmed Mr. Biden’s victory. There is no evidence of widespread election fraud in Georgia or other states.

At the rally, Mr. Trump called the January elections “the most important congressional runoff probably in American history” and said Georgia voters would decide which party controls every Senate committee, writes legislation and directs taxpayer dollars. Victories by Democrats Jon Ossoff, a documentary filmmaker opposing Sen. David Perdue, and Raphael Warnock, facing Sen. Kelly Loeffler, would leave each party with 50 seats in the Senate and give Democrats control, given that the Vice President-elect Kamala Harris can break ties.

The president invited Mr. Perdue and Ms. Loeffler onto the stage. Both heaped praise on the president, eliciting a round of “Fight for Trump” chants.

The president alleged that the voting system was plagued by widespread fraud but urged voters to turn out for next month’s elections. The crowd repeatedly chanted “Stop the steal.”

The president’s comments came hours after he called Georgia Gov. Brian Kemp, a Republican, to push for the state legislature to reverse Mr. Biden’s win in the state, according to a person familiar with the call, part of the president’s unprecedented effort to overturn his election loss.

Mr. Kemp declined the president’s request, the person familiar with the conversation said. Mr. Trump later repeated the request at the rally, which Mr. Kemp didn’t attend.

In his phone call with Mr. Kemp, Mr. Trump asked the governor to call a special session of the state legislature and to push lawmakers to appoint pro-Trump electors who would swing the Electoral College in his favor, the person familiar with the call said. The Washington Post first reported details of Mr. Trump’s request.

Mr. Biden’s narrow win in Georgia, the first for a Democratic presidential candidate since 1992, sparked excitement among Democrats and frustration among Republicans, including some Trump allies who have grumbled that the president’s campaign ignored pleas for more money to fund get-out-the-vote efforts in the state in the final stretch before Election Day. The campaign and Republican National Committee denied rejecting any requests.

”Georgia Democrats know how much is on the line in this Jan. 5 election,” said Democratic state Rep. Dexter Sharper, who represents the Valdosta area. Electing Messrs. Ossoff and Warnock, he said at a press conference, would help the Biden administration rein in the coronavirus and “pass real economic relief for small businesses and working families.’’

Throughout the rally, the president alleged that illegal voting had taken place in November. He took several swipes at Mr. Kemp and Georgia’s Republican secretary of state, Brad Raffensperger, whom he has repeatedly attacked in recent weeks.

The crowd booed Mr. Kemp at various points. Mr. Trump asked Rep. Doug Collins (R., Ga.) if he would consider running for governor in two years.

“You’ve got to make sure your secretary of state knows what the hell he’s doing, and you’ve got to make sure your governor gets a lot tougher,” Mr. Trump said. He accused both men of being afraid of Stacey Abrams, a popular Democratic politician in the state who narrowly lost the 2018 gubernatorial campaign to Mr. Kemp.

Georgia Republicans have grown increasingly alarmed about the effect that Mr. Trump and his allies may have on the Senate race with their efforts to overturn Mr. Biden’s victory. Of particular concern is the message pro-Trump lawyers L. Lin Wood and Sidney Powell have been sending to the state’s GOP voters, urging them not to vote in the runoffs and promoting an unsupported allegation that the state’s voting machines were rigged and must be fixed.

At a “Stop the Steal” protest on Wednesday in the Atlanta suburb of Alpharetta, Mr. Wood also criticized Mr. Perdue and Ms. Loeffler. “They have not earned your vote. Don’t you give it to them,” Mr. Wood said. “Why would you go back and vote in another rigged election, for God’s sake?”

Supporters who attended Saturday’s rally said they distrusted the election process and worried that similar feelings would lead other Republicans to skip voting in the Senate runoffs.

“We worried about the voting, but we’re going to show up,” said Arica Harris, a 36-year-old insurance agent from Valdosta. “Otherwise, what are we going to do, let them win by a landslide?”

Many in the crowd were from South Georgia, but others came over the border from nearby Florida, with some coming from as far away as New York and Missouri.

Republicans have dominated Georgia for decades and today hold every statewide office, as well as a majority in both chambers of the legislature. In recent years, however, Democrats have been gaining, due in large part to the migration of younger people and minorities to the state. Organizing efforts by Democrats, including those by Ms. Abrams, have also helped the party.

Ahead of the runoff elections, both sides have flooded television, radio and streaming music services with ads. Text messages are being sent to cellphones. The intensity of the campaigns here has at least matched the presidential contest.