He told Fox News he had not made a "final, final decision" on a running mate. He heaped praise on Pence and his other two finalists.
Viewed as a safe pair of hands, Pence, 57, has diverging views with Trump on his proposed Muslim ban and trade, and is more socially conservative, but he could help unify a divided party behind Trump's White House bid.
Trump was due to make his official announcement on his choice on Friday at 11 a.m. (1500 GMT) in Manhattan. But he tweeted on Thursday night that the attack in Nice, where a truck slammed into a crowd, killing dozens of people, prompted him to delay.
"In light of the horrible attack in Nice, France, I have postponed tomorrow's news conference concerning my Vice Presidential announcement," said Trump. He said in a Fox News interview: "We will announce tomorrow when it will be."
Trump, who has proposed banning Muslims from "terror states" from entering the United States, said in another Fox News interview that the attack in France showed the United States and the rest of the world needed to get tougher in the fight against Islamist militants.
"This has to be dealt with very harshly," Trump said.
He told Fox News he had not made a "final, final decision" on a running mate. He heaped praise on Pence and his other two finalists, former House of Representatives Speaker Newt Gingrich and New Jersey Governor Chris Christie.
"I've got three people that are fantastic," he said.
Trump's advisers told national party officials that he had settled on Pence, according to two Republican sources familiar with the campaign's operations.
"I'm told he's been asked to do this and he's flying to New York," one source said. Pence was seen by TV networks arriving at a New York-area airport.
Trump, 70, a New York businessman, is to be formally nominated as the party's candidate for the Nov. 8 presidential election at the Republican National Convention next week in Cleveland. Traditionally, the vice presidential choice is used to build enthusiasm among party loyalists.