"In the letter, the president mentioned topics of interest that Senator Paul wanted to discuss with President Putin", spokesman Hogan Gidley said in a statement.
Among the proposals Putin's Trump - extension for five years of validity of the contract on Strategic offensive arms (start-III).
The June 12 meeting between Trump and North Korean leader Kim Jong-Un was successful 28 percent of respondents said, while 33 percent said it was unsuccessful and 39 percent said they were not sure.
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Paul has been in Russia this week and has met with Russian lawmakers in a trip he sees as a continuation of Trump's diplomatic outreach to Putin.
FILE PHOTO: U.S. President Donald Trump receives a football from Russian President Vladimir Putin as they hold a joint news conference after their meeting in Helsinki, Finland, July 16.
"We expect that in the nearest time it will come to the presidential administration", he said, according to Russian news agencies. However, he undermined those remarks when he quickly returned to calling the special counsel's investigation a "witch hunt", and referred to Russian interference as a "hoax".
In addition, Putin in Helsinki allegedly spoke with trump about the new ban the deployment of weapons in outer space and called for the establishment of a consultative mechanism at the level of Ministers of defence and foreign Affairs. "He is interested in legislative exchanges, parliamentary dialogue and cultural and educational exchange programs, increased counter-terrorism cooperation, building on recent successes like the disruption of the St. Petersburg plot and resolution of the military conflicts in Syria and Ukraine".
Republican Senate Majority Leader Mitch McConnell last week identified the bill as a potential step Congress could take to push back against Russian Federation as Senate Democratic leader Chuck Schumer called for sanctions and other deterrents.