(Adds background, rouble, Kremlin comment on letter)
WASHINGTON, Aug 8 (Reuters) - Republican U.S. Senator Rand Paul said on Wednesday he delivered a letter from President Donald Trump to Russian President Vladimir Putin proposing cooperation, weeks after Trump was widely condemned for not standing up to Putin at their Helsinki summit.
Trump has pushed for easing tensions between the former Cold War foes, exacerbated by U.S. intelligence findings that Russia meddled in the 2016 presidential election, trying to tilt the outcome to Trump.
Last week, Trump's national security team said Russia is behind "pervasive" attempts to interfere in the November midterm elections, in a rejection of denials of meddling that Putin made directly to Trump. Trump sparked outrage by accepting Putin's word over that of his intelligence agencies at a Helsinki news conference.
Paul, one of the few Republicans to defend Trump's July 16 performance in Helsinki, said he had traveled to the Russian capital to encourage diplomacy amid tense relations between Moscow and Washington.
"I was honored to deliver a letter from President Trump to President Vladimir Putins administration. The letter emphasized the importance of further engagement in various areas including countering terrorism, enhancing legislative dialog and resuming cultural exchanges," Paul wrote in a Twitter post.
The White House had no immediate comment on the letter.
Kremlin spokesman Dmitry Peskov said on Wednesday that a letter from Trump to Putin had been delivered by diplomatic channels, the Interfax news agency reported. But Peskov said the Kremlin had not familiarized itself with the contents.
Paul, part of the party's conservative wing who frequently takes antiwar positions, on Monday invited Russian lawmakers to visit Washington to discuss nuclear nonproliferation and terrorism. He also held talks in Moscow with parliamentarians and pledged to block new sanctions against Russia.
The two countries are also at odds over Syria and Ukraine.
Further U.S. sanctions on Russia over alleged election meddling, Syria and Ukraine could lead to a deeper rift and are hurting Russia's rouble.
Paul's colleagues in the Senate from both parties introduced legislation last week to impose strict new sanctions on Russia. A reported text of the draft bill published on Wednesday sent Russia's rouble to lows against the dollar last seen in April. (Reporting by Doina Chiacu, Lisa Lambert and Jeff Mason Editing by Mary Milliken and Jonathan Oatis)