As Senate runoffs approach, Trump’s attacks on Georgia Republicans have worried some in his party.

President Trump’s sustained assault on his own party in Georgia, and his repeated claims of election fraud in the state, have intensified worries among Republicans that he could be hurting their ability to win two crucial Senate runoff races next month.

The president has continued to claim without evidence that his loss in the new battleground state was fraudulent, directing his ire in particular at Gov. Brian Kemp and Secretary of State Brad Raffensperger, both conservative Republicans, whom he has accused of not doing enough to help him overturn the result.

Over the weekend, he escalated his attacks on Mr. Kemp, saying he was “ashamed” to have endorsed him in 2018, and on Monday he called Mr. Kemp “hapless” on Twitter as he urged him to “overrule his obstinate Republican Secretary of state.’’

Mr. Trump’s broadsides have quietly rattled some Republicans in the state, who fear that concerns about the fairness of the presidential election could depress turnout for the Senate races, which will determine whether Democrats or Republicans control the chamber.

After resisting entreaties to appear in Georgia, the president plans to travel there this weekend, though even some of his own aides remain uncertain whether his anger toward state officials will overshadow any support he may lend the party’s two candidates.

“You can’t say the system is rigged but elect these two senators,” said Eric Johnson, a former Republican leader of the Georgia Senate who is a campaign adviser to Kelly Loeffler, one of the Republican candidates. “At some point he either drops it or he says I want everybody to vote and get their friends to vote so that the margins are so large that they can’t steal it.”

The split signifies both an extraordinary dispute over election integrity within the Republican Party and a preview of the control the president may continue to exert over the conservative base even after he leaves office. As Mr. Trump talks seriously about the possibility of mounting another bid for the White House in 2024, his personal goals may not always align with those of his party — no matter the political stakes.

“I had someone message me just last week saying: ‘Nope, I’m done. Can’t trust the election. Never voting again,’” said Buzz Brockway, a former Republican state representative. “The president has a very dedicated group of supporters who don’t really support the broader Republican Party — they support him.”