Mr. Clapper said that bringing Mr. Putin, a former K.G.B. chief, into the White House would pose stiff intelligence risks. “This will be a complex intelligence and counterintelligence challenge,” he said.

Mr. Coats also said he was not fully aware of what Mr. Trump and Mr. Putin discussed in their one-on-one meeting on Monday in Helsinki, Finland, but that he hopes to learn soon, a remarkable admission for a cabinet-level national security official.

And asked whether Mr. Trump should avoid a similar one-on-one meeting with Mr. Putin if he comes to the White House, Mr. Coats said he would “look for a different way of doing it.” White House officials said on Thursday that Mr. Trump planned to invite Mr. Putin to visit in the fall.

Mr. Coats also said, with a measure of diplomacy, that he had not been aware of the 2017 meeting in the Oval Office between Mr. Trump and Sergey V. Lavrov, Russia’s foreign minister, along with Sergey I. Kislyak, then the Russian ambassador to the United States. During their discussion, Mr. Trump revealed sensitive Israeli intelligence.

That meeting, Mr. Coats said, was “probably not the best thing.”

Mr. Coats’s appearance at Aspen after a week of controversy was itself something of a departure, given Mr. Trump’s avowed dismissal of establishment politics. The security forum is a gathering of foreign policy experts that some Trump supporters view as the playground of the so-called deep state, the kind of establishment national security experts fiercely opposed by the current White House.

Earlier in the day, Kirstjen Nielsen, the secretary of homeland security, faced questions defending the administration’s immigration policies, which are deeply unpopular with much of the crowd at Aspen.

In contrast, Mr. Coats’s tough line on Russia won him strong applause. Still, it risks the ire of the White House. For all the care Mr. Coats, an experienced politician, has taken to hedge his criticism of Mr. Trump, the gap between the president’s views and his has never been more clear or in the open.