Donald Trump's claim that thousands of Muslim New Jersey residents were cheering after the Sept. 11, 2001, terrorist attacks led to a vivid back-and-forth as Trump critics challenged the real estate tycoon's account.

The former "Apprentice" star insisted on Tuesday he would be happy to apologize for any erroneous statement, but in this particular case, he has no doubts about his recollection, CBS News noted.

"There were a lot of happy people over in New Jersey," the 69-year-old Republican presidential candidate reiterated as he spoke to a crowd at a campaign rally in the crucial early primary state of New Hampshire.

"I saw it, and a lot of people saw it, and I'm getting hundreds of phone calls and a lot of other people are too, and things are all of a sudden materializing," Trump said. "So people said, 'You should apologize.' I said. 'I don't apologize. Hey look, if I'm wrong about something, I want to apologize.'"

Trump also pointed to a 2001 newspaper article, and his campaign adviser cited clips from WCBS and MTV he said substantiated his account. But neither video showed a large celebration, CBS detailed.

Former Republican presidential candidate Rudy Giuliani, who served as New York mayor's during the time of the 2001 attacks, meanwhile, warned Donald Trump he might have a huge political problem on his hands if he were not able to produce any evidence about Muslim Americans' celebrations, The Guardian reported.

"I don't want to say he's not telling the truth or not. Let him deal with it. Let him explain to people. Let him show the evidence of it," Giuliani told CNN on Tuesday.

"If thousands of people were demonstrating and he saw it on television, then there must be some tape of it somewhere. If it shows up, it will corroborate him. If it doesn't show up, it's going to make him look really bad."