The White House raised hopes Monday of achieving a breakthrough agreement to resolve bitter differences with Moscow over missile defenses in Europe when President Bush meets with Russian President Vladimir Putin this weekend.

Bush's national security adviser, Stephen Hadley, said no deal was in hand yet but the two leaders could nail it down when they meet Sunday in the Black Sea resort of Sochi. "We may. We're hopeful," he said. It will be the last meeting between the two men before Putin steps away from the Russian presidency.

Hadley briefed reporters on Air Force One as Bush flew to Ukraine to begin a weeklong trip in Eastern Europe, anchored by a NATO summit in Bucharest, Romania. Arriving in Kiev late at night, Bush was presented with a traditional greeting of bread and salt. His wife, Laura, was given a bouquet of flowers.

The Western military alliance has been strained by the refusal of Germany and other allies to send more combat troops to Afghanistan, prompting accusations from Washington that they are shirking their duty. France announced last week it would send more forces, probably a battalion of elite paratroopers. That has reduced some of the pressure and allowed Bush and other leaders to step back from a NATO clash. Britain and Poland also are expected to do more.

Defense Secretary Robert Gates, in Denmark ahead of the NATO summit, set measured expectations.

"I would be surprised if we saw commitments in Bucharest at a level that would fully meet all the requirements" for combat troops and military and police trainers, Gates said. "But we'll just keep working at it."

The United States wants not only more troops, but also fewer restrictions from some governments on how their troops can be used.

"We've all been saying that we all need to do more," Hadley said. "We've also been saying this is going to be a long effort and we're going to have to be committed to a long-term effort in Afghanistan. I think that's true. We need to step it up. I think you'll find that countries are stepping up. That's a good next step. But there are going to be more steps down the road."

The U.S. proposal for a missile defense shield in Eastern Europe has been a major source of friction between Washington and Moscow.

For months, Putin has ratcheted up his anti-American rhetoric, demanding that the United States abandon the plan, which would be based in Poland and the Czech Republic, two former Soviet satellites. Putin has complained it would upset the balance of power and was aimed at weakening Russia, charges the United States has repeatedly denied.

In recent days, there have been signs of progress toward resolving the dispute. Bush sent Gates and Secretary of State Condoleezza Rice to Moscow with concessions to ease Russia's concerns. Bush also sent a personal letter to Putin. A Russian delegation spent several days in Washington last week working on the problem.

The United States has offered to let Russia monitor the system and share in the information that is collected. Bush also has offered not to activate the system until there is a verifiable threat from Iran or some other adversary.

"I think we're moving in a direction where ... Russia and the United States could have missile defense as an area of strategic cooperation," Hadley said.

Bush, in an interview last week, insisted the missile shield was not aimed at Russia. "After all, it doesn't take many missiles to overwhelm the kind of system we're talking about," Bush said. "And Russia has got plenty of missiles if they want to overwhelm." Bush said the shield was intended to protect from missiles launched from the Middle East, where the United States regards Iran as a primary threat.

Before traveling to Romania, Bush stopped in Ukraine to praise its democratic reforms and encourage its drive to join NATO. Ukraine wants to be put on the path toward eventual membership and hopes NATO will provide a membership action plan that outlines what it needs to do to join. Georgia also wants a membership action plan, which is a precursor to the granting of full membership.

Russia is adamantly opposed to either Romania or Georgia getting on the NATO track. With nine former Soviet bloc countries already in NATO, Russia fiercely opposes the eastward expansion of an alliance it denounces as a Cold War relic.

Germany and France have spoken out against putting Ukraine on the list just yet, fearing upsetting already strained ties with Russia, a major supplier of energy to Europe.

NATO is expected to formally invite Croatia and Albania to join the alliance. Macedonia also is on the list but could be blocked by Greece. Greece has insisted it will veto Macedonia if it does not change its name. Greece feels the name, which is the same as a neighboring province of northern Greece, implies a territorial claim.