MOSCOW—Sen. Rand Paul met with Russian lawmakers Monday and invited them to Washington as part of a growing outreach to the Kremlin, days after a bipartisan group of U.S. senators proposed tightening sanctions against Russia over findings of election interference.

“Russia shouldn’t be considered our friend, but we have certain shared interests—regarding Syria, terrorism and energy,” said Mr. Paul (R., Ky.) after meeting with lawmakers from the Federation Council, the upper house of parliament.

The...