Washington (CNN) Donald Trump twice repeated over the weekend that he saw people cheering in New Jersey after the September 11 attacks -- but his claims are being widely disputed as false.

GOP primary rival Ben Carson also said he witnessed the same, but his campaign walked back his statement later on Monday.

Trump first made the claim at a campaign rally in Alabama on Saturday, where he said he "watched the World Trade Center go down" and watched in New Jersey, "as thousands of people were cheering as the building was coming down."

He doubled down on those claims in an interview Sunday with ABC's "This Week," even as George Stephanopoulos pushed back.

"There were people that were cheering on the other side of New Jersey where you have large Arab populations," Trump said. "They were cheering as the World Trade Center came down. I know it might be not politically correct for you to talk about it, but there were people cheering as that building came down."

"Police say it didn't happen," Stephanopoulos pressed, to which Trump said he saw it himself on television.

Trump claimed: "It was well covered at the time."

But numerous publications and local politicians have said Trump is incorrect.

Neither outlet could find any news reports that corroborate Trump's account of events. There were a few reports of rumors that there were celebrations in New Jersey, along with images of Palestinians celebrating overseas that were broadcast locally.

A September 17, 2001, an Associated Press story referred to "unfounded ... rumors of rooftop celebrations of the attack by Muslims" in Jersey City, New Jersey.

And in a September 18, 2001, article, The Washington Post wrote that "law enforcement authorities detained and questioned a number of people who were allegedly seen celebrating the attacks and holding tailgate-style parties on rooftops while they watched the devastation on the other side of the river." But there was no source for the information and there is no evidence of anything coming of the questioning.

Politicians from the area also pushed back.

"Either @realDonaldTrump has memory issues or willfully distorts the truth, either of which should be concerning for the Republican Party," Democratic Jersey City Mayor Steven Fulop tweeted Sunday.

Either @realDonaldTrump has memory issues or willfully distorts the truth, either of which should be concerning for the Republican Party — Steven Fulop (@StevenFulop) November 22, 2015

"Not sure what luxury spider-hole @realDonaldTrump was hiding in on Sept11 but I saw Americans come together that day @GStephanopoulos," tweeted former New York Gov. George Pataki -- a competing candidate in the GOP primary.

Not sure what luxury spider-hole @realDonaldTrump was hiding in on Sept11 but I saw Americans come together that day @GStephanopoulos — George E. Pataki (@GovernorPataki) November 22, 2015

"That is totally false. That is patently false," Jerry Speziale, the police commissioner of Paterson, New Jersey, told The Washington Post. "That never happened. There were no flags burning, no one was dancing. That is (barnyard epithet)."

One of Trump's GOP primary opponents, New Jersey Gov. Chris Christie, was noncommittal on whether it happened, but said he didn't recall anything of the sort.

"I do not remember that, and so it's not something that was part of my recollection. I think if it had happened, I would remember it, but, you know, there could be things I forget, too," Christie told NJ.com

Trump initially did get back up from Carson, who told reporters gathered in Nevada that he did see celebrations of American Muslims in New Jersey after 9/11.

"I saw the film of it, yes," he said. Asked what kind of film, he said: "The news reels."

"You know there are going to be people who respond inappropriately to virtually everything," Carson said. "I think that was an inappropriate response. I don't know if on the basis of that you can say all Muslims are bad people. I really think that would be a stretch."

Carson's campaign walked back that comment later, and then he did so himself in an interview with Fox News host Megyn Kelly. Carson said he was speaking about Muslims celebrating 9/11 -- but not about their exact location.

"What we were talking about is the reaction of Muslims after the 9/11 attack and if they were in a celebratory mood. And, you know, I was really focusing on that it was an inappropriate thing to do, no matter where they were," Carson said.

He stood by his comment that he'd seen films of the celebrations, but said, "I don't know where they were."

Carson also accused reporters of mischaracterizing his remarks.

"I thought we were talking about the fact that Muslims were inappropriately celebrating. I didn't know that they had an agenda behind the question," he said.

The reporter had asked: "Were American Muslims in New Jersey cheering on 9/11, when the towers fell -- did you hear about that or see that?" Carson answered, "Yes."

Florida Sen. Marco Rubio denounced Trump's claim Monday after a campaign event in Iowa.

"It's not true and there's plenty of fact-checks to prove that it isn't," he told reporters there, according to his campaign.

Democratic presidential candidate Bernie Sanders told CNN's Brooke Baldwin on Monday that he didn't know where Trump was getting his evidence.

"I don't think anybody else in America has seen it," Sanders said. "What I get concerned about, Brooke, is the growth of Islamophobia in this country, the desire to win votes by scapegoating a group of people, which is not what America is supposed to be about. So I think once again, Mr. Trump is missing the boat."

Trump himself also demanded an apology from The Washington Post later Monday linking to the 2001 article referencing murky reports of allegations of celebrating.

"Via @washingtonpost 9/18/01. I want an apology! Many people have tweeted that I am right!," Trump tweeted

Via @washingtonpost 9/18/01. I want an apology! Many people have tweeted that I am right! https://t.co/CpsMxs54qv pic.twitter.com/wrDEhXJlvR — Donald J. Trump (@realDonaldTrump) November 23, 2015

Serge Kovaleski, the lead reporter on the Washington Post story that Trump links to, said he doesn't remember witnesses reporting large groups of people celebrating.

"We did a lot of shoe leather reporting in and around Jersey City and talked to a lot of residents and officials for the broader story. Much of that has, indeed, faded from memory," said Kovaleski, who's now an investigative reporter for The New York Times. "But I do not recall anyone saying there were thousands, or even hundreds, of people celebrating. That was not the case, as best as I can remember."

The Washington Post had already updated its story many hours earlier to include the quote from its 2001 story and said it didn't change the rating it was giving Trump's claims.

The comments came during a weekend where Trump called for surveillance of mosques, the use of waterboarding, a registry list for all refugees and in which he retweeted a follower's tweet containing bogus statistics falsely implying blacks were responsible for most murders of both races.