Former CIA Director John Brennan continued Tuesday to criticize President Trump’s performance at the Helsinki summit, saying Mr. Trump could have potentially risked intelligence capabilities during his private meeting with Russian President Vladimir Putin and the intel community might begin to withhold intelligence from Mr. Trump.

In light of Mr. Trump’s comments about the intelligence community, Mr. Brennan said “there very well might be” a shift in the covert community to “withhold vital intelligence” in order to protect that information.

“Dan Coats and Gina Haspel need to be on top of their game now and speak truth to power,” he said, referring to Director of National Intelligence Dan Coats and CIA Director Gina Haspel. “And make sure Trump doesn’t get away with this.”

Mr. Brennan said Mr. Trump was “way, way out of his depth” meeting one-on-one with the “master manipulator,” Mr. Putin.

No CIA agent would have entertained an individual private talk with any Russian, Mr. Brennan said, because “that’s just something we don’t do.”

The former CIA director warned that Mr. Trump could have potentially compromised or impacted intelligence capabilities during that private meeting.

“I still don’t understand why he didn’t trust John Bolton, Mike Pompeo to be in that meeting,” Mr. Brennan said.

Mr. Brennan said it would be “unbelievable” if Russia did not keep a single recording of Mr. Trump’s private statements, but he wasn’t sure the U.S. did.

“I believe that whatever Mr. Trump said in that meeting is now memorialized on Russian tape and will be utilized by Putin against Trump,” the former CIA director said.

On Monday, Mr. Brennan accused Mr. Trump’s comments at the joint press conference of being “nothing short of treasonous.”

The next day, Mr. Brennan criticized the president for siding with Mr. Putin and throwing the U.S. intelligence community “under the bus” rather than doing what was “minimally acceptable” by addressing the indictments.

Mr. Brennan suggested the Russians do have dirt on the president. He said Mr. Trump’s behavior “may reflect that concern” that the Russians will use his past against him.

The former head spy refused to say he believed that anything in particular or tangible, like a tape as some have suggested, exists.

“I’m just saying Trump knows and that’s why he’s so desperate to stop the Mueller investigation,” Mr. Brennan said, “Because clearly he’s concerned and, I think, fearful about what might be exposed in that investigation.”

Mr. Trump responded and told Fox News that Mr. Brennan “is a very bad guy.”

“If you look at it, a lot of things happened under his watch [leading the CIA under then-President Barack Obama]. I think he is a very bad person,” Mr. Trump told host Tucker Carlson in an interview airing Tuesday night.

• Dave Boyer contributed to this article.