Paul, part of the party's conservative wing who frequently takes antiwar positions, on Monday invited Russian lawmakers to visit Washington to discuss nuclear nonproliferation and terrorism.
CNN is airing a series of panels with Trump voters from the 2016 election, some of whom stand by the president and others who now regret their vote.
"The letter emphasized the importance of further engagement in various areas including countering terrorism, enhancing legislative dialogue and resuming cultural exchanges", Paul said on Twitter.
U.S. President Donald Trump receives a football from Russian President Vladimir Putin as they hold a joint news conference after their meeting in Helsinki, Finland, July 16.
Details in the single page agenda for the meeting shows Mr Putin remains interested in maintaining continued cooperation with the U.S. on nuclear weapons.
"What bothers me more is that I think he's a suck-up to Vladimir Putin because he has something on him", he said.More news: Oil rises as US renew sanctions against Iran
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Paul has emerged as one of Trump's most vocal allies on Russian Federation, even as the president's warmer rhetoric, including trying to cast doubt on Moscow's 2016 election interference, has sparked bipartisan backlash on Capitol Hill.
Paul, an ally of Trump, said he had traveled to the Russian capital to encourage diplomacy amid tense relations between Moscow and Washington.
According to Russian state news agency RIA Novosti, Putin's press secretary Dmitry Peskov confirmed that the letter was sent to Putin "through diplomatic channels", but the Kremlin has not yet reviewed it.
Damani Bryant Felder, a Trump supporter, pointed out that given how prolifically leaky the administration is, if the Russians had something on Trump we would know by now.
The two countries are also at odds over Syria and Ukraine.