President Donald Trump and Russian President Vladimir Putin shake hands as they hold a joint news conference after their meeting in Helsinki, Finland, July 16, 2018.

President Donald Trump sent a letter to Russian President Vladimir Putin recently, emphasizing "the importance of further engagement in ... countering terrorism, enhancing legislative dialogue and resuming cultural exchanges."

This was according to Sen. Rand Paul, R-Ky., who delivered the letter to Putin's administration during a visit to Russia this week. The letter comes less than a month after Trump's heavily criticized performance at a joint press conference with Putin in Helsinki.

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The letter also came just days after both Russia and China stopped just short of admitting they intend to ignore the reimposed U.S. sanctions on Iran that went into effect Monday, and to continue doing business with the country. Trump had threatened that "anyone doing business with Iran will NOT be doing business with the United States."

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The White House said the letter was Paul's idea, not Trump's. "At Senator Paul's request, President Trump provided a letter of introduction" to Putin, deputy press secretary Hogan Gidley said in a statement Wednesday. "In the letter, the President mentioned topics of interest that Senator Paul wanted to discuss with President Putin."

The White House did not respond to a specific question from CNBC about what the term "cultural exchanges" meant in this context.

But another one of the goals identified in the letter, increased "legislative dialogue," already appears to be underway. During the past month, eight Republican senators have traveled to Moscow and met with Russian officials and members of the Duma, Russia's parliament.

A group of seven GOP senators traveled to Moscow in early July, and upon their return, said they forcefully raised the issue of Russia's attack on the 2016 U.S. presidential election, as well as a host of other destabilizing activities the Kremlin has undertaken around the world.

Paul traveled alone, however. Like Trump, Paul takes a far more conciliatory stance toward Putin's government than do many U.S. intelligence agencies, legislators or cybersecurity experts.

In a statement Tuesday, Paul said he had invited a delegation of Russian legislators to Washington "to continue dialogue on vital issues such as nuclear non-proliferation and combating terrorism," and his invitation had been accepted.

Trump sent a similar invitation to Putin following the Helsinki summit, but plans for any such visit are still on hold.

In July, special counsel Robert Mueller indicted 13 Russians for allegedly hacking into the computers of Democratic campaigns and officials in 2016. Dozens more Russian government officials are also currently subject to U.S. sanctions stemming from Russia's 2014 annexation of Crimea.

This story has been updated to include a statement issued by the White House.