• In an interview, Mr DeSantis said "stay tuned" on his rumoured presidential ambitions, while taking a shot at his fellow Republican's legal woes.
• He levelled veiled criticism of the "underlying conduct" behind Mr Trump's alleged affair with a porn star.
Florida Governor Ron DeSantis has reportedly taken a swipe at Donald Trump's "daily drama", sharpening his attacks on a top White House rival.
In an interview, Mr DeSantis said "stay tuned" on his rumoured presidential ambitions, while taking a shot at his fellow Republican's legal woes.
He levelled veiled criticism of the "underlying conduct" behind Mr Trump's alleged affair with a porn star.
The rivals have been shadow-boxing ahead of the 2024 White House race.
On Tuesday, Trump allies slammed Mr DeSantis' latest remarks. Most Republican party figures have voiced support for the legally embattled Mr Trump.
A grand jury in New York is currently considering whether to recommend charges against the former president over a 2016 payment to an adult film actress to keep quiet about their alleged affair.
In an interview with Piers Morgan Uncensored that airs on Thursday, Mr DeSantis vowed that if he runs for the White House he could defeat President Joe Biden.
Previewing his interview, the Fox Nation host said that when he asked Mr DeSantis to contrast his style with Mr Trump's, the Florida governor said: "The way we run the government I think is no daily drama, focus on the big picture and put points on the board."
Asked about Mr Trump's nicknames for him, including "Ron DeSanctimonious" and "Meatball Ron", he said: "To me, it's just background noise.
"It's not important for me to be fighting with people on social media. It's not accomplishing anything for the people I represent."
On the topic of a former Trump fixer's hush money payment to porn star Stormy Daniels, Mr DeSantis said there had been "a lot of speculation about what the underlying conduct is".
"That is purported to be it, and the reality is that's just outside my wheelhouse. I mean that's just not something that I can speak to."
On Monday Mr DeSantis angered Trump supporters by saying: "I don't know what goes into paying hush money to a porn star to secure silence over some type of alleged affair."
On Tuesday night, a spokesman for Mr Trump said the latest criticism proved that Mr DeSantis has "terrible political instincts".
"While the entire conservative movement is united against the unjust indictment of President Donald Trump, Governor DeSantis is choosing to go off half-cocked and take shots on some low-rent vlog," tweeted Taylor Budowich.
Donald Trump Jr, the former president's eldest son, accused Mr DeSantis of "pathetically" attacking Mr Trump "on orders" from the Republican "establishment".