Prosecuting Donald Trump would be a politically motivated move by his opponents, senior Republicans say.
On Saturday, the former US president said his arrest could happen on Tuesday, based on media reports which say an indictment could come next week.
Prosecutors in New York have not commented on the progress of the probe.
The case focuses on alleged hush money paid on Mr Trump's behalf by his lawyer to porn star Stormy Daniels ahead of the 2016 presidential election.
The 76-year-old is also the subject of several other separate inquiries, although he has not yet been charged in any and denies wrongdoing.
Mr Trump has pledged to continue his campaign to become the Republican nominee in the 2024 presidential election, even if he is indicted.
It is not yet known if he is going to be criminally charged or even, beyond the most basic details, what any indictment might contain.
Former US vice-president Mike Pence and the most senior Republican in the US Congress, Kevin McCarthy, have both spoken out against any criminal prosecution.
Speaking to US network ABC News, Mr Pence said Manhattan district attorney Alvin Bragg, who is a Democrat, was carrying out a "politically charged prosecution" against Mr Trump.
"I'm taken aback at the idea of indicting a former president of the United States, at a time when there's a crime wave in New York City," Mr Pence said.
"The fact that the Manhattan DA [district attorney] thinks that indicting President Trump is his top priority, I think is, just tells you everything you need to know about the radical left in this country," he told the broadcaster.
In his statement on Saturday Mr Trump called for mass protests from supporters and accused Mr Bragg's office of "illegal leaks" to journalists.
Asked about the call for protests, Mr Pence said that US citizens have a "constitutional right to peaceably assemble" - but also said any action must take place "peacefully and in a lawful manner".
Speaker of the US House of Representatives Kevin McCarthy called the investigation "an outrageous abuse of power by a radical DA".
In a tweet, he also promised to investigate whether federal money was being used to interfere in elections "with politically motivated prosecutions".
Democrats accused Mr Trump of stirring up political divisions with his claims.
Speaking to ABC News, Democratic Senator Elizbeth Warren said "no-one is above the law" even if they happen to have been president and any investigation must be "allowed to go forward appropriately".
"There's no reason to protest this. This is the law operating as it should without fear or favour for anyone," she said.
Mr McCarthy's predecessor as House Speaker, Nancy Pelosi, echoed Ms Warren's sentiments - adding that Mr Trump's remarks were "reckless" and designed to "keep himself in the news and to foment unrest among his supporters".
"He cannot hide from his violations of the law, disrespect for our elections and incitements to violence. Rightfully, our legal system will decide how to hold him accountable," she wrote on Twitter.
The Stormy Daniels case centres on how Mr Trump reimbursed his lawyer Michael Cohen after he paid Ms Daniels $130,000 before the 2016 US election.
The record for the payment reimbursing Mr Cohen says it was for "legal fees". Prosecutors could conclude this amounts to Mr Trump falsifying business records, which is a misdemeanour offence in New York.