Russian President Vladimir Putin at a meeting on 16 July in Helsinki gave the American leader is Donald Trump papers with a proposal to extend for five years the Treaty on the limitation of nuclear arsenals.
"The letter emphasised the importance of further engagement in various areas including countering terrorism, enhancing legislative dialogue and resuming cultural exchanges", Mr Paul wrote in a Twitter post.
Paul's Moscow trip, Trump said the junior senator from Kentucky "would appreciate the opportunity to meet with you to discuss several topics".
Undaunted by the backlash in his own party to his first meeting, Trump invited Putin to a White House meeting sometime this autumn.
Fox News chief White House correspondent John Roberts revealed the contents of the letter on Wednesday.
US President Donald Trump and the key members of his team have been repeatedly rejecting all the accusations concerning collusion between the Trump campaign and Russian Federation.
The White House has neither confirmed nor denied the authenticity of the document, except to state that President Trump did not receive any written suggestions from Putin in Helsinki.
"The world is a complicated place, we are in close proximity to Russian Federation in Syria and other places, and I think it would be a very big mistake not to have open lines of communication", Paul said on Monday.
Details in the single page agenda for the meeting shows Mr Putin remains interested in maintaining continued cooperation with the United States on nuclear weapons.
Christopher J. Gagin, who revealed he was the Republican Party chairman from OH who resigned after the Helsinki summit, pointed out that Trump defended Putin's denials of Russian meddling in the 2016 election.
The senator met with members of the Russian parliament's upper house who serve on the foreign affairs committee Monday.
Congressman Massie suggested in an interview with Breitbart News that America and the world will benefit from the increased dialogue between the two countries.