A grand jury was expected to indict Trump late on Monday or on Wednesday, Politico reported, for arranging payments to porn star Stormy Daniels.
Trump, who is seeking the Republican nomination for the White House again in 2024, had predicted he would be arrested on Tuesday.
But on Monday, the grand jury still appeared to be collecting evidence before deciding whether charges were warranted.
New York Mayor Eric Adams told reporters police were monitoring social media and keeping an eye out for "inappropriate actions" in the city.
Trump faces two civil trials involving former magazine columnist E. Jean Carroll, who claims that Trump defamed her by denying he raped her.