WASHINGTON — A conservative operative trumpeting his close ties to the National Rifle Association and Russia told a Trump campaign adviser last year that he could arrange a back-channel meeting between Donald J. Trump and Vladimir V. Putin, the Russian president, according to an email sent to the Trump campaign.

A May 2016 email to the campaign adviser, Rick Dearborn, bore the subject line “Kremlin Connection.” In it, the N.R.A. member said he wanted the advice of Mr. Dearborn and Senator Jeff Sessions of Alabama, then a foreign policy adviser to Mr. Trump and Mr. Dearborn’s longtime boss, about how to proceed in connecting the two leaders.

Russia, he wrote, was “quietly but actively seeking a dialogue with the U.S.” and would attempt to use the N.R.A.’s annual convention in Louisville, Ky., to make “ ‘first contact.’ ” The email, which was among a trove of campaign-related documents turned over to investigators on Capitol Hill, was described in detail to The New York Times.

Robert S. Mueller III, the special counsel investigating Russian interference in the election and possible collusion with the Trump campaign, secured a guilty plea on Friday from President Trump’s first national security adviser, Michael T. Flynn, for lying to the F.B.I. about contacts with Moscow’s former ambassador to the United States. But those contacts came after Mr. Trump’s improbable election victory.