But the Americans desperately wanted the hacking tools. The cyberweapons had been built to break into the computer networks of Russia, China and other rival powers. Instead, they ended up in the hands of a mysterious group calling itself the Shadow Brokers, which has since provided hackers with tools that infected millions of computers around the world, crippling hospitals, factories and businesses.

No officials wanted to refuse information they thought might help determine what had happened.

“That’s one of the bedeviling things about counterintelligence and the wilderness that it is — nobody wants to be caught in a position of saying we wrote that off and then five years later saying, ‘Holy cow, it was actually a real guy,’” Mr. Hall said.

American intelligence agencies believe that Russia’s spy services see the deep political divisions in the United States as a fresh opportunity to inflame partisan tensions. Russian hackers are targeting American voting databases ahead of the midterm election this year, they said, and using bot armies to promote partisan causes on social media. The Russians are also particularly eager to cast doubt on the federal and congressional investigations into the Russian meddling, American intelligence officials said.

Part of that effort, the officials said, appears to be trying to spread information that hews closely to unsubstantiated reports about Mr. Trump’s dealings in Russia, including the purported video, whose existence Mr. Trump has repeatedly dismissed.

Rumors that Russian intelligence possesses the video surfaced more than a year ago in an explosive and unverified dossier compiled by a former British spy and paid for by Democrats. Since then, at least four Russians with espionage and underworld connections have appeared in Central and Eastern Europe, offering to sell kompromat to American political operatives, private investigators and spies that would corroborate the dossier, American and European intelligence officials said.

American officials suspect that at least some of the sellers are working for Russia’s spy services.

The Times obtained four of the documents that the Russian in Germany tried to pass to American intelligence (The Times did not pay for the material). All are purported to be Russian intelligence reports, and each focuses on associates of Mr. Trump. Carter Page, the former campaign adviser who has been the focus of F.B.I. investigators, features in one; Robert and Rebekah Mercer, the billionaire Republican donors, in another.

Yet all four appear to be drawn almost entirely from news reports, not secret intelligence. They all also contain stylistic and grammatical usages not typically seen in Russian intelligence reports, said Yuri Shvets, a former K.G.B. officer who spent years as a spy in Washington before immigrating to the United States after the end of the Cold War.