According to the newspaper, "the White House is drafting an executive order that would authorize President Trump to sanction foreigners who interfere in USA elections, the administration's latest effort to demonstrate it is serious about combating Russian disinformation and hacking".
Paul, a Republican, said on Twitter that "the letter emphasized the importance of further engagement in various areas including countering terrorism, enhancing legislative dialogue and resuming cultural exchanges".
After the summit, Paul and Trump discussed the senator's planned meetings with Russian officials and the messages the president wanted Paul to convey to them.
Paul, who defended Trump in the wake of his summit with Putin last month in Helsinki, has been visiting Russian Federation with a delegation for several days.
One White House official emphasized that the document is a draft and that it could change but said it showed Trump is intent on safeguarding USA elections.
Representative Trey Gowdy, the Republican chairman of the House Oversight and Government Reform Committee, questioned the wisdom of Putin being ushered into the White House.
The White House later released its own statement underscoring that the letter was "introductory" and written at the request of Paul.
The president has called special counsel Robert Mueller's investigation into possible ties between his campaign and Moscow a "witch hunt".
The president later corrected some of his remarks, saying he misspoke.
A United States official provided to CNN the text of the letter, which read: "I would like to introduce Senator Rand Paul of Kentucky, a member of the Senate Foreign Relations Committee, and a voice for expanding dialogue with the Russian Federation.
He also held talks in Moscow with parliamentarians and pledged to block new sanctions against Russian Federation. In addition to tensions over election meddling, the two countries are also at odds over Syria and Ukraine. The U.S. director of national intelligence would be required to conclude if any foreign nations interfered in elections one month after Americans cast their votes, triggering strict sanctions within 10 days if interference was detected.