WASHINGTON — John O. Brennan, the former director of the Central Intelligence Agency, said Wednesday that he thought Russia may have some kind of compromising information on President Trump, setting off furious speculation about whether the former spy chief was basing that assertion on inside information.

In an appearance on Wednesday on MSNBC’s “Morning Joe,” Mr. Brennan, who served as C.I.A. director from 2013 to 2017, speculated that Russians “may have something on him personally” when he was asked if he thought Mr. Trump was afraid of President Vladimir V. Putin of Russia.

“The Russians, I think, have had long experience with Mr. Trump, and may have things that they could expose,” he added.

Mr. Brennan was running the C.I.A. when a salacious dossier surfaced in 2016 that claimed the Russians had compromising information on Mr. Trump. If there were any current or former American officials who might know if there was truth behind the allegations in the dossier, Mr. Brennan would most likely be one of them. And his comments came the day after a phone call Mr. Trump made to Mr. Putin congratulating him for winning an election raised new questions about the president’s relationship with Russia.