Meanwhile, Mr. DeSantis needs to decide whether to joust more frequently with Mr. Trump or try to avoid engaging directly with a foe who thrives on such confrontations.

His strategy has mostly been to keep the focus on the Biden administration rather than Mr. Trump, people close to him say. Responding to the personal attacks, they say, only fuels the media attention Mr. Trump commands. Some outside observers are skeptical of that approach.

“I think for DeSantis, the time to engage is getting very close,” said Terry Sullivan, who managed Sen. Marco Rubio’s 2016 presidential bid. “We all made the mistake in 2016 thinking it would be easy to take Trump on, one on one. So we wasted our time fighting for second.”