Paul tweeted that the letter "emphasized the importance of further engagement in various areas including countering terrorism, enhancing legislative dialogue and resuming cultural exchanges".
Both Trump and Putin have invited each other to their respective capital cities in the aftermath of last month's Helsinki summit between the two leaders.
"I was honored to deliver a letter from President Trump to President Vladimir Putin's administration", Paul wrote.
Topics the pair negotiated during their private summit in Helsinki, Finland on July 16 included the Russian president's willingness to extend a series of landmark nuclear treaties and pursue new weapons limits, according to the leaked document.
The three Trump supporters said they were comfortable with the widely panned display from the US president, and proceeded to clash with the dissenters.
The White House again refused to answer questions about what was taken about during the summit but would neither confirm or deny the validity of the document.
"My goal in coming to Russian Federation is to say that we want to have open lines of communication", Paul said at a press conference after the meeting.
Paul's Moscow trip, Trump said the junior senator from Kentucky "would appreciate the opportunity to meet with you to discuss several topics".
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Putin's spokesman, Dmitry Peskov, said the Kremlin has not yet examined the letter Paul presented.
Last week, Trump's national security team said Russian Federation is behind "pervasive" attempts to interfere in the November midterm elections, rejecting denials that Putin made directly to Trump.
The Kentucky Republican met with members of the Russian parliament's upper house who serve on the foreign affairs committee Monday.
Paul remained supportive of Trump following the US President's performance during a news conference with Putin in July.
He also held talks in Moscow with parliamentarians and pledged to block new sanctions against Russian Federation.
Further U.S. sanctions on Russian Federation over alleged election meddling, Syria and Ukraine could lead to a deeper rift and are hurting Russia's rouble.
The two countries are also at odds over Syria and Ukraine.
A reported text of the draft bill published Wednesday sent Russia's ruble to lows against the dollar last seen in April.