“The more he talks about this in terms of not being sure, the more he throws our intelligence communities under the bus, the more he’s willing to forgive and forget Putin, the more suspicion,” Mr. Graham added in an interview on NBC’s “Meet the Press.” “And I think it’s going to dog his presidency until he breaks this cycle.”

As if to underscore the point, the White House confronted reports later Sunday that Donald Trump Jr., Mr. Trump’s eldest son, was promised damaging information about Hillary Clinton before agreeing to meet with a Kremlin-connected Russian lawyer at Trump Tower during the campaign last year. The accounts of the meeting, by three White House advisers briefed on it and two others with knowledge of it, represent the first public indication that at least some people in Mr. Trump’s campaign were willing to accept Russian help.

Reince Priebus, the White House chief of staff, had played down that meeting during an appearance on “Fox News Sunday,” calling it a “nothing meeting,” and a “big nothing burger.”

President Trump’s account of his lengthy and closely scrutinized closed-door meeting with Mr. Putin on the sidelines of the Group of 20 summit meeting came in a series of Twitter posts the morning after he had returned from the gathering in Hamburg, Germany. They appeared to be an attempt to move beyond the controversy after Moscow characterized the election discussion as a meeting of minds rather than a showdown between the American president and his Russian counterpart.

Administration officials knew that Mr. Trump’s much-anticipated meeting with Mr. Putin was risky and in some ways a no-win situation. The tangle of investigations into his campaign’s possible dealings with Russia raised the stakes and created a damaging backdrop for Mr. Trump, while Mr. Putin’s well-earned reputation for outfoxing and manipulating adversaries suggested that he would stage manage the meeting for maximum advantage, making himself appear to have the upper hand.