“When you are stealing secrets, and those guys are talking about the Trump administration, then those guys are going to be in a tricky spot,” said Daniel Hoffman, a former C.I.A. operations officer who served in Moscow.

Still, Mr. Hoffman disputed the idea that the charged atmosphere would create a morale crisis in the C.I.A. He said that even during the early years of the Iraq war, when many intelligence officers fought with the White House over the intensity of the insurgency there, the functions of spycraft continued.

“We recruited spies. We stole secrets. We did the work,” he said.

Mr. Trump has been at odds with most of the national security establishment since the beginning of his administration. He and his allies view members of the intelligence agencies as part of a so-called deep state opposed to his policies.

Mr. Trump had been planning to ask Mr. Putin to the White House since their Helsinki meeting, two people familiar with the event said. But bringing Mr. Putin, a former K.G.B. chief, into the White House would pose stiff security risks, said James R. Clapper Jr., a former director of national intelligence.

“The Russians will be leaning forward to both collect intelligence, and to thwart what we do to collect against them,” Mr. Clapper said. “Similarly, we will be leaning forward to collect against them, and to thwart their efforts to collect against us.”

Some intelligence officials reacted with resignation to Mr. Trump’s plan. While previous presidents would have consulted them about the risks of such a meeting, the officials have become increasingly convinced that Mr. Trump is not fully absorbing their briefs, even when they are tailored to his tastes with models, physical demonstrations and extensive use of photographs.

Intelligence officials are growing concerned that Mr. Trump cherry-picks their findings to reinforce decisions he has already made, several administration officials said in interviews. They noted that in the case of North Korea, he picked up on evidence last summer of growing nuclear capabilities to bolster his threats of military action; now that he is pursuing a thaw in relations with North Korea, he is ignoring similar evidence.