CNN host Alisyn Camerota led off with Trump's "performance" in Helsinki at his summit with Russian President Vladimir Putin.
Paul 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".
The meeting last month was also attended by the Russian Foreign Ministry and a U.S. delegation led by Thomas Pickering, former Undersecretary of State.
The two had savaged each other on the 2016 presidential primary campaign trail, with Paul at one point deriding Trump as an "orange-faced windbag" and the president retorting that Paul was "truly weird".
Paul has been in Russian Federation this week on a trip he sees as a continuation of Trump's diplomatic outreach to Putin.
Paul, an ally of Trump, said he had traveled to the Russian capital to encourage diplomacy amid tense relations between Moscow and Washington.
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.
The White House downplayed the significance of the letter on Wednesday, insisting that it was merely a formal introduction requested by Paul and laying out the "topics of interest that Senator Paul wanted to discuss with President Putin", deputy press secretary Hogan Gidley said. But Peskov said the Kremlin had not familiarized itself with the contents.
On Monday, Sen. Paul announced that Russian lawmakers have agreed to visit the U.S. Capitol, which would be the first time in three years that a Russian delegation did so.
Congressman Massie suggested in an interview with Breitbart News that America and the world will benefit from the increased dialogue between the two countries.
Senators are mulling legislation to slap new sanctions on Russian Federation. A reported text of the draft bill published on Wednesday sent Russia's rouble to lows against the dollar last seen in April.
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