Paul said that members of the Foreign Affairs Committee of the upper house of the Russian parliament, the Federation Council, had been invited "to come to the United States to meet with us in Washington".
"Engagement is vital to our national security and peace around the world".
Kosachev said Paul "has access" to Trump and top USA officials "and we expect that we will be able to convey through him our signals" of a desire to establish dialogue, according to the state-run RIA Novosti news service.
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Paul is in Moscow meeting with Russian lawmakers in a trip he sees as a continuation of US President Donald Trump's diplomatic outreach to Russian President Vladimir Putin, and comes several weeks after Trump invited Putin to DC as well.
Speaking after the meeting that lasted about an hour, Paul said it had been "wonderful" and announced he had invited members of the Russian committee to meet with their counterparts in the USA from the Senate Foreign Relations Committee, on which he sits.
Paul's trip to Russian Federation was first announced in mid-July.
Paul headed a USA delegation, which includes member of the Texas State Senate Don Huffines and representatives of the Cato Institute, a public policy research organization. Trump was heavily criticized by both Republicans and Democrats for being too deferential to Putin in his remarks and appearing to side with him over USA intelligence agencies' assessment of Russian interference in the 2016 election.
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"They use the anti-Russian campaign purely for internal American purposes, the interests of the same elections campaign they are charging us with meddling in", Kosachev said.
And Paul is the only Republican lawmaker to stand by Trump during that controversy.
One reporter asked if the senator had discussed election interference with the Russian officials.
Putin said he would be happy to meet Trump in Washington or Moscow, but the White House has since said the meeting will be postponed to 2019.
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Prime Minister Shinzo Abe , who also was at the ceremony, reiterated Japan's pledge to maintain its pacifist and non-nuclear principles.