Paul said the two "agreed on the importance of continued dialogue".
The senator was one of Trump's most vocal supporters in Washington last month when the USA president came under fire for his handling of a summit with President Vladimir Putin.
Sen. Paul met with several Russian senators during his trip to Russia on Monday, as well as Sergey Kislyak, Russia's former envoy to Washington.
Kosachev heads up the Russian equivalent of the U.S. Senate Foreign Relations Committee, which Paul sits on.
Sen. Paul said he visited to show Russians that there are "many Americans who want to have diplomacy, that want to have engagement".
The Kentucky Republican was in Moscow meeting with Russian officials on Monday.
The U.S. visit will be the first in three years for the. "Meanwhile, my Trump-defending opponent Don Huffines made a decision to go on a Moscow adventure this week with Rand Paul instead, to "improve parliamentary relations" with Russian Federation".
The invitation comes at a time of heightened tensions between the USA and Moscow, with lawmakers pushing legislation that would levy additional sanctions on Russian Federation for election meddling.
Shelby's delegation said it had come with the similar goal of reviving communications with Russia, but had also delivered a warning to the Russians not to interfere in the US mid-terms this November.
Trump endured days of intense bipartisan criticism for his performance at a joint press conference, where he cited Putin's "powerful" denials of Russian interference. "It does not mean we don't have differences".
It was "opposition research", Trump said, not collusion.
Mr. Paul's press released noted that nuclear proliferation and terrorism will be among the issues the worldwide leaders discuss.
Mr. Paul was the only lawmaker on the trip, which was financed by the Cato Institute, a leading libertarian research organization in Washington.
"We had general discussions about a lot of issues and basically we've decided that right now we will try to is establish a dialogue and solve issues".
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