The $US4000 ($A6000) total penalty prosecutors are seeking would come on top of a $US9000 fine Justice Juan Merchan imposed on Tuesday, when he held the Republican presidential candidate in contempt for social media posts that questioned the jury selection process and insulted his former lawyer Michael Cohen, who will be a crucial witness.
Merchan said he might jail Trump if he continued to defy the gag order, saying the fines allowed by New York law - $US1000 per violation - might not be enough to deter for the wealthy businessman-turned-politician.
The gag order aims to prevent Trump intimidating witnesses, jurors and other participants in the first criminal trial of a former US president.
It does not prevent Trump criticising prosecutors or the judge himself.
Trump says the gag order restricts his free-speech rights and prevents him responding to political attacks.
On Tuesday, he repeated his claim that prosecutors were working with Democratic President Joe Biden to undercut his bid to win back the White House.
He also repeated his claim that Merchan faced a conflict of interest because his daughter had done work for Democratic politicians.
Trump is accused of falsifying business records to hide a hush-money payment to porn star Stormy Daniels shortly before the 2016 presidential election.
Lawyer Keith Davidson testified on Tuesday that Daniels had been shopping her story of a 2006 sexual encounter with Trump to media outlets at a time when Trump was already facing damaging accusations of sexual misbehaviour.
Trump has pleaded not guilty and says he did not have sex with Daniels, whose real name is Stephanie Clifford.
Before testimony resumes on Thursday, Merchan will consider whether Trump violated the gag order on four occasions last week by referring to Cohen as a "liar" and former National Enquirer publisher David Pecker, another witness, as a "nice guy" in statements to news media.
Prosecutors say Trump also violated the gag order by saying in a television interview that "that jury was picked so fast - 95 per cent Democrats. The area's mostly all Democrat."
Trump faces three other criminal prosecutions, though it is not clear whether any of them will go to trial before the November 5 presidential election.
Two accuse him of trying to overturn his 2020 election loss to Biden, while another accuses him of mishandling classified documents after leaving office.
He has pleaded not guilty in all three cases.
His legal troubles have come at a cost.
Fundraising groups have diverted tens of millions of dollars from his presidential campaign to his legal fees, and he has had to post $US266 million in bonds in order to appeal two civil judgments that found he engaged in business fraud and defamed writer E Jean Carroll, who claimed he raped her in the 1990s.
