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 meetings marked the second U.S. delegation to Moscow led by a Republican congressman this summer and gave Russian officials a chance to promote their position on a number of issues that strain already tense relations between Washington and Moscow, from interference in elections to cyber security.

“We will continue this dialogue,” Mr. Paul said on Russian state television, adding that a visit would likely happen by the end of the year.

Following President Trump’s summit with Vladimir Putin in Helsinki, the U.S. president invited his Russian counterpart to Washington, though the trip has been pushed back from the fall to sometime next year. A previous Republican delegation to Moscow last month included Sen. Richard Shelby, Sen. John Kennedy and Rep. Kay Granger.