Donald Trump wasn't backing down Monday on his controversial claim that he saw "thousands" of people cheering in Jersey City, N.J., "when the World Trade Center came tumbling down" on Sept. 11, 2001 -- even as the city's mayor said the front-running Republican presidential hopeful has "memory issues, or willfully distorts the truth."

Trump first told the story Saturday at a rally in Birmingham, Ala., as he pressed the need for greater surveillance, including monitoring certain mosques, in the wake of the Paris attacks.

"I watched when the World Trade Center came tumbling down. And I watched in Jersey City, New Jersey, where thousands and thousands of people were cheering as that building was coming down," Trump said.

Trump repeated the claim Sunday in an interview on ABC's "This Week" after host George Stephanopoulos explained that police had refuted any such rumors at the time.

"It did happen. I saw it," said Trump. "It was on television. I saw it."

"There were people that were cheering on the other side of New Jersey, where you have large Arab populations. They were cheering as the World Trade Center came down," he said.

"I know it might be not politically correct for you to talk about it," he added, "but there were people cheering as that building came down, as those buildings came down. And that tells you something."

A spokeswoman did not respond to a request for clarification Saturday about Trump's comments.

In a statement, Jersey City Mayor Steven Fulop criticized Trump for his remarks.

"Trump is plain wrong, and he is shamefully politicizing an emotionally charged issue," said Fulop. "No one in Jersey City cheered on September 11th. We were actually among the first to provide responders to help in lower Manhattan." 

The Anti-Defamation League, in a statement to BuzzFeed, called the comments "irresponsible" and a "variation of the anti-Semitic myth that a group of Israelis were seen celebrating as the Twin Towers fell.” 

Footage of Muslims in Middle Eastern countries cheering news of the attacks were broadcast often on television, but there is no evidence in news archives of mass celebrations by Muslims in Jersey City, which sits right across the Hudson River from Lower Manhattan, with clear views of the World Trade Center site.

While rumors have circulated on the internet for years that American Muslims celebrated the attacks in Paterson, New Jersey, police officials and religious leaders denied it at the time.

"Trump needs to understand that Jersey City will not be part of his hate campaign," said Fulop. "Clearly, Trump has memory issues or willfully distorts the truth, either of which should be concerning for the Republican Party."

The Associated Press contributed to this report.