U.S. President George W. Bush and Russia's Vladimir Putin began a final effort on Sunday to mend frayed relations face-to-face but with little hope of resolving the biggest dispute that divides them.

In a farewell summit at the Russian president's Black Sea holiday villa, the two leaders - both in the twilight of their terms - will use their personal chemistry to try to bridge differences that have driven ties to a post-Cold War low.

Their two countries are bitterly divided over a U.S. plan for a missile defence shield in eastern Europe and on the eve of Sunday's talks in the resort of Sochi, the White House all but ruled out any deal this weekend.

Opening formal talks in a guesthouse at Putin's villa complex, Bush stressed the personal rapport the two men have built in more than 20 meetings over seven years.

"This is the final meeting we'll have as presidents of our respective countries," Bush said.

"I've come to respect you ... you've been a strong leader, you're not afraid to tell me what's on your mind. And when it's all said and done, we can shake hands."

Bush also held his first substantive talks with Putin's protege and successor, Dmitry Medvedev, who will be sworn in as president on May 7.

"After taking over the post of Russian President, I want to make it so that our relations continue further without pauses, in a constructive spirit," Medvedev told Bush.

"For eight years you and President Putin have done much to develop U.S. and Russian relations. These relations are a key factor in global security," he said.

Bush told Medvedev: "I'm looking forward to getting to know you so we'll be able to work through common problems and find common opportunities."

Bush flew to Sochi after a NATO summit in Bucharest which tackled issues including the missile shield and the alliance's enlargement along Russia's borders, both of which are a source of concern to the Kremlin.

DANCING PRESIDENTS

The summit opened on a positive note after Bush's arrival on Saturday when the two joined in a Russian folk dance during an informal dinner of red caviar and veal loin.

The missile shield is chief among the difficult issues that include Kosovo, Iran and Russia's record on human rights.

Despite the tensions, Bush hopes to capitalise on a less strident tone struck by Putin at the NATO summit last week, where he attacked Western military expansion near Russia's borders but also implored: "Let's be friends, guys."

The two men are expected to sign an agreement on the "strategic framework" of U.S.-Russian relations, aimed at encapsulating the legacy they will bequeath to their successors.

Bush, who leaves office in January, was meeting the Russian leader seven years after saying he had peered into Putin's soul and trusted him. Critics had roundly mocked Bush as naive.

Showing their rapport was still intact, the two leaders shared a hug as Bush arrived at Putin's compound. They exchanged jokes as they were briefed on plans for the 2014 Winter Olympic Games in Sochi, Putin's pet project.

Bush is out to salvage a foreign policy record dominated by the Iraq war, which has damaged U.S. credibility.

He is also struggling to stay relevant on the world stage now that his economic stewardship is under fire and attention is focused on whoever will succeed him in the White House.

Putin is looking for a smooth transition of the presidency to Medvedev, elected in a landslide last month after the Russian leader endorsed him.