The former C.I.A. director John Brennan pulled no punches on Wednesday when he was asked why President Trump had congratulated his Russian counterpart, Vladimir Putin, for his victory in a rigged election, even after Mr. Trump’s national security staff warned him not to.

“I think he’s afraid of the president of Russia,” Mr. Brennan said, on MSNBC’s “Morning Joe” program, of the phone call on Tuesday between the two presidents. “The Russians, I think, have had long experience with Mr. Trump and may have things they could expose.”

The possibility that Mr. Putin could have some hold on the American president has lurked in the background over the past year as Mr. Trump displayed a mystifying affection for the Russian leader and ignored or excused his aggressive behavior and nefarious activities, most important, his interference in the 2016 campaign, a subject of the special counsel Robert Mueller’s investigation.

Some Trump defenders noted that President Barack Obama also called Mr. Putin when he was elected president in 2012.