Though intelligence officials have disputed that the officer who delivered the main briefing said Russia was actively aiding the president’s re-election, people in the room said that intelligence officers’ responses to lawmakers’ follow-up questions made clear that Russia was trying to get Mr. Trump re-elected.

Intelligence is hardly a perfect process, as Americans learned when the nation went to war in Iraq based in part on an estimate that Saddam Hussein was once again in search of a nuclear weapon.

Not surprisingly, the Kremlin says this is all an American fantasy, aimed at demonizing Russia for the United States’ own failings. “These are more paranoid announcements which, to our regret, will multiply as we get closer to the election,” Mr. Putin’s confidant and spokesman, Dmitry S. Peskov, was quoted by Reuters as telling reporters on Friday. “They have nothing to do with the truth.”

No matter who is elected, Mr. Putin has probably undermined one of his own primary goals: getting the United States and its allies to lift sanctions that were imposed after he annexed Crimea and accelerated a hybrid war against Ukraine.

“By actively exploiting divisions within American society and having its activities revealed, the Kremlin has ensured that its longer-term goal of having the U.S. remove sanctions and return to a less confrontational relationship so far has been thwarted,” Angela E. Stent, a former national intelligence officer for Russia and now a professor at Georgetown University, said in her book “Putin’s World: Russia Against the West and With the Rest.”

Ms. Stent noted on Saturday that if the Russians were in fact interfering in this election, “it could bring about new energy sanctions.” She said that one piece of legislation in the Senate would require new sanctions if evidence of Russian meddling emerged from intelligence agencies. So far, Mr. Putin may have concluded that the penalties are a small price to pay if he can bring his geopolitical rival down a few more notches, she added. And the early intelligence analyses suggested that by backing Mr. Sanders in the primary and Mr. Trump in the general election, Mr. Putin would probably have a good chance of maximizing the electoral tumult.

Robert O’Brien, the president’s national security adviser, said in an interview with ABC to be broadcast on Sunday that he had not seen evidence that Russia sought to intervene in Mr. Trump’s favor, and that the reports of Mr. Putin helping Mr. Sanders’s campaign came as “no surprise.”