Paul said that members of the Foreign Affairs Committee of the upper house of the Russian parliament, the Federation Council, had been invited "to come to the United States to meet with us in Washington".
The chairman of the Russian worldwide relations committee, Konstantin Kosachev, thanked Paul and said he hoped the meetings could happen before the end of the year.
The Republican senator and ally of U.S. President Donald Trump said he had traveled to the Russian capital to encourage diplomacy amid tense relations between Moscow and Washington.
Both Republican and Democratic lawmakers back in Washington quickly pointed out in tweets and formal statements that Russian Federation is not a friend of the US and that the president should not accept Putin's word as bond on matters of USA national security. Republicans spoke to their foreign counterparts in July, ahead of the Helsinki summit, to discuss several issues including Russian meddling in USA elections.
There is unanimous agreement among US intelligence agencies that the Russian government directed extensive influence operations to sow racial, political, and cultural division in the election.
Background: A group of USA lawmakers led by Sen.
Paul, however, has stood by Trump, despite his remarks in Helsinki.More news: Rand Paul Delivers Trump Letter To Putin
More news: Saudi Arabia withdrawing students from Canada
More news: The Pentagon bans the use of mobile location services for security reasons
Trump has repeatedly said he wants to improve relations with Russian Federation, which are severely strained.
State Sen. Don Huffines, R-Dallas, met with Russian officials in Moscow on Monday, joining U.S. Sen.
"I am one who believes in more engagement, that we need to have more cultural exchange, more exchange between our legislative bodies, more open lines of communication", Paul said.
Mr. Paul was the only lawmaker on the trip, which was financed by the Cato Institute, a leading libertarian research organization in Washington.
Trump's Helsinki meeting was panned by a number of prominent Republican officials who said the president appeared to side with Putin over top US intelligence officials. Republican Senator Lindsey Graham told reporters earlier in the week that he wanted "a sanctions bill from hell".
"We had general discussions about a lot of issues and basically we've decided that, right now, what we're trying to do is have dialogue", he said. "So we're not going to get into any of the differences other than we are trying to agree to have dialogue".
Senior Russian lawmaker Konstantin Kosachev, who was also in the room, reiterated the Russian position that "there was no interference in 2016", adding "there of course will not be any interference in the elections this year".