SOCHI, Russia (Reuters) - President George W. Bush and Russia’s Vladimir Putin will have one last chance on Sunday to try to mend frayed relations face-to-face but with little hope of resolving the biggest dispute that divides them.

President George W. Bush (L) and his Russian counterpart Vladimir Putin watch the sunset at the Bocharov Ruchei residence at the Black Sea in Sochi April 5, 2008. REUTERS/RIA Novosti/Kremlin

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.

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

The summit opened on a positive note after Bush’s arrival at Putin’s seaside retreat on Saturday as the two joined in a Russian folk dance during an informal dinner of red caviar and veal loin. But once the night’s festive mood has worn off, the leaders still face a long list of grievances.

The thorniest issue is Bush’s plan for a U.S. missile shield based in eastern Europe, a system Washington says is needed to protect against Iran but Moscow says threatens its security.

After holding out hope a deal on missile defense might be reached at Sochi, the White House all but ruled that out on the eve of Sunday’s formal talks.

“We’re going to have to do more work after Sochi,” White House spokeswoman Dana Perino said on the Bush’s Air Force One presidential jet as it headed to Russia. She called it “premature” to predict a deal though negotiations over compromise proposals were “headed in the right direction”.

Despite that, Bush hopes to capitalize on a less strident tone struck by Putin at a NATO summit in Bucharest 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.

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“HEART-TO-HEART”

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 naïve.

Since then, disputes over Kosovo, missile defense and Russia’s rights records have led to talk of a Cold War-style atmosphere.

Bush says he expects his last heads-of-state meeting with Putin -- they have met more than 20 times before -- to be another “heart-to-heart.”

Showing their rapport was still intact, the two leaders shared a hug as Bush arrived at Putin’s compound and 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.

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

Bush had a chance to start forging a bond with Medvedev over dinner, and will hold separate talks with him on Sunday.

This may give Bush a chance to gauge how much power Putin, who will become prime minister, will continue to wield.

In the meantime, some Russia experts have suggested Medvedev may keep Bush at arm’s length, just as Putin brushed off Bill Clinton in 2000 in the final months of his presidency.

Also certain to figure prominently in Sunday’s talks will be NATO’s eastward expansion. As a guest of honor at the NATO summit in Bucharest, Putin stopped short of the searing attacks he had launched on Washington and its allies over the past year.

NATO leaders rebuffed Bush’s call to put Georgia and Ukraine on an immediate path to membership, a move some Europeans feared would have deeply antagonized Russia.

But Moscow made clear its concern about the alliance’s ultimate decision -- a promise to the two former Soviet republics that they would eventually be allowed in.