Mr. Napolitano, who keeps a residence in Manhattan at Trump International Hotel & Tower on Central Park West, did not respond to inquiries on Friday.

But Mr. Johnson, who was himself once a Fox News contributor, said in a telephone interview that Mr. Napolitano called him on Friday and requested that he speak to The New York Times. Mr. Johnson said he was one of the sources for Mr. Napolitano’s claim about British intelligence.

Mr. Johnson became infamous in political circles after he spread false rumors in 2008 that Michelle Obama had been videotaped using a slur against Caucasians. In the interview on Friday, Mr. Johnson acknowledged his notoriety, but said that his knowledge of surveillance of Mr. Trump came from sources in the American intelligence community. Mr. Napolitano, he said, heard about his information through an intermediary.

“It sounds like a Frederick Forsythe novel,” Mr. Johnson said.