The leader of a national Muslim-American organization characterized Donald Trump's unsubstantiated claim that he saw Jersey City residents celebrating on the day of the 2001 terrorist attacks as "vicious" and invited the Republican presidential candidate to come to the city and meet with Muslims.

Nihad Awad, executive director of the Council on American-Islamic Relations, said Thursday that Trump has produced no evidence to back up his claim last month that he saw television reports of "thousands of people" in Jersey City cheering as the Twin Towers fell Sept. 11, 2001.

Trump also has said he would use a database to track Muslims in the U.S. if he's elected. And Awad said Trump should be held "at least partly responsible" for an uptick in hostility toward Muslims in the U.S. because of his anti-Muslim rhetoric.

Both Republican Gov. Chris Christie, a presidential candidate, and Jersey City Mayor Steven Fulop, a Democrat, say Trump's claim is wrong.

"The fact is people can say whatever they like," Christie said Monday while campaigning. "But the facts are the facts, and that did not happen in New Jersey that day, and it hasn't happened since."

Trump's campaign didn't immediately respond to a request for comment Thursday.