Given Mr. Perdue’s ties with Mr. Trump, it is possible that the former president may be a presence campaigning for Mr. Perdue and against Mr. Kemp next year.

Still, it’s not entirely clear that a Republican Senate candidate should welcome Mr. Trump’s future assistance.

Bill Crane, a Georgia political operative and commentator, said on Monday that the former president’s appearances on behalf of the two Republicans appeared to have worked against them in January — with Republican turnout depressed in the two congressional districts where Mr. Trump campaigned.

Mr. Crane, who has worked for both Republican and Democratic candidates, said he would not be surprised if Mr. Perdue runs against Mr. Warnock given the close results in his January race, adding that, to win, Mr. Perdue would have to change his strategy.