“He heard the comments, took some time to digest them and decided to go out and make a defense of the intelligence community,” said Shawn Turner, a former spokesman for the Office of the Director of National Intelligence in the Obama administration.

The contretemps prompted speculation in Washington that Mr. Coats could step down. Such a decision is far from clear, said Michael V. Hayden, who ran the C.I.A. and National Security Agency under President George W. Bush and has emerged as a critic of Mr. Trump.

“It’s a tough choice for him: Am I helping America by staying or am I helping more by leaving? That’s now a meaningful question that he is going to have to answer,” Mr. Hayden said.

Mr. Turner said Mr. Coats demonstrated political courage by speaking out, particularly given Mr. Trump’s record of attacking those in his administration he views as disloyal. “It is more important than ever for officials to stand their ground and speak out when the president does something counter to our beliefs,” Mr. Turner said.

Trump administration supporters said too much was being made of the president’s comments at the news conference with Mr. Putin, pointing to his clarification. Mr. Trump said his remarks were a misunderstanding that arose when he stumbled over a “double negative.”

But some Trump allies viewed Mr. Coats’s recent comments suspiciously. They said they believe Mr. Coats has been too lenient toward the so-called deep state and view him as an obstacle to Mr. Trump’s goal of overhauling the intelligence agencies to make them smaller and more efficient. These officials said they believe Mr. Coats is too quick to trust intelligence officers who may be pursuing an agenda separate from the administration’s.

Mr. Coats served as a Republican in the Senate originally from 1989 to 1999, first appointed to fill Dan Quayle’s seat when Mr. Quayle became vice president. A social conservative, Mr. Coats was credited with helping develop the compassionate conservative agenda embraced by George W. Bush in the 2000 campaign.