The senator posted about the meeting on his official Facebook page.
"Over the last few weeks, I've crisscrossed the district talking to constituents about the many challenges we face right here in Senate District 16", Johnson wrote.
Paul said he and Kosachev agreed "engagement is vital to our national security and peace around the world, adding that "We agreed on the importance of continued dialogue".
Russian foreign affairs chairman Konstantin Kosachev says that no plans on meetings have been finalized, but that he hoped such a meeting was possible to arrange before the end of the year.
The delegation with Paul included Phillip Huffines, a Texas Republican who ran unsuccessfully in a primary race this spring, and Peter Goettler, president and CEO of the Cato Institute, the conservative Washington-based think tank.
It was "opposition research", Trump said, not collusion.
NASA announces crew for commercial spaceships
SpaceX has also faced its own problems, particularly with its Falcon 9 rocket that will power the Dragon into space. It now looks like SpaceX will be the first to launch humans on its crew Dragon spacecraft, set for April 2019.
Paul, who has emerged as a defender of President TrumpDonald John TrumpSouth Korea urges Pyongyang to speed up denuclearization process More than a dozen arrested as protesters, counter-protesters clash in Berkeley Trump golfs with Graham at New Jersey club MORE's rhetoric toward Russian Federation, announced last month that he planned to visit Moscow in an effort to improve diplomatic relations between the two countries.
"It does not mean that we have no differences, it does not mean that we will not, on occasion, have discussions back and forth where we disagree".
"I think this is incredibly important", the United States lawmaker was quoted as saying in translated comments by Interfax news agency.
"My goal in coming to Russian Federation is to say that we want to have open lines of communication, it does not mean that we have no differences, it does not mean that we will not, on occasion, have discussions back and forth where we disagree", Paul said.
This week, both Democratic and Republican senators introduced a bill that would impose the harshest sanctions yet on Moscow as punishment for its USA election interference, as well as its activities in Ukraine and Syria.
Kosachev said members of his committee are "ready to discuss this question in a reasoned manner" with USA counterparts. "They're not going to admit it in the same way we're not going to admit we were involved in the Ukrainian elections or the Russian elections", he said.