MOSCOW—Russian President Vladimir Putin said the re-election of President Barack Obama could improve relations with the U.S., but that he was also prepared to work with Mitt Romney, calling the Republican candidate's tough stance on Russia "pre-election rhetoric."

In contrast to what had been viewed as a chilly attitude toward Mr. Obama, Mr. Putin called his U.S. counterpart "a genuine person" who "really wants to change much for the better." Speaking to Russia's state-run RT television channel, he...