While the president was in Paris last week, Mr. DeSantis let the White House know, through intermediaries, that Mr. Trump’s incendiary tweets accusing Democrats of trying to steal the election were hurting and not helping, according to Republicans and administration officials with knowledge of the situation.

The difference may be not only in style, but in mission: Mr. Scott believes he is preparing to stride into the fractious partisan battleground of the United States Senate. Mr. DeSantis is preparing to govern the nation’s third-largest state — Mr. Gillum conceded the race Saturday after a recount.

Breaking with the most powerful Republicans in his state, Mr. DeSantis has grown frustrated with the bombastic attacks on Democrats launched by Mr. Trump, Mr. Scott and Senator Marco Rubio, believing it will erode confidence in elections and spark a Democratic backlash, according to two Florida Republicans who have spoken with him in recent days.