However, deputy White House spokesman Hogan Gidley said the missive was a "letter of introduction" that Trump had written at Paul's request and "mentioned topics of interest that Senator Paul wanted to discuss with President Putin".
Paul has been one of the president's most vocal defenders in recent weeks, defending Trump for appearing to accept Putin's denials of USA election interference.
The Kentucky Republican says Russian and American lawmakers need to work closer than before, saying the lack of dialogue is the biggest problem between the two countries.
The White House told Fox News that Paul asked Trump to write a letter of introduction to the Russian leader.
At the time, uncertainty about what Mr Trump and Mr Putin discussed during the private meeting in which members of the press were also banished from caused a stir.
Trump soon after his summit with Putin had invited the Russian president to Washington this fall, but Trump's national security adviser, John Bolton, later said any meeting would be put off until 2019.
Paul, one of the few Republicans to defend Trump's widely panned performance at a July 16 news conference with Putin in Helsinki, said he had traveled to Moscow to encourage diplomacy.
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.
Kremlin spokesman Dmitry Peskov said that a letter from Trump to Putin had been delivered by diplomatic channels, the Interfax news agency reported. Paul said in a statement on Wednesday.
The White House pushed back against Sen. In a joint news conference with Trump, Putin said he did want Trump to win because then-candidate Trump had said he wanted to improve U.S.
Paul, a member of the Senate Foreign Relations Committee, has continued to meet with Russian officials in Moscow this week along with visiting the US embassy.
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.
Paul's colleagues in the Senate from both parties introduced legislation last week to impose strict new sanctions on Russian Federation.