Still, harsh differences divide Moscow and Washington, particularly on security issues.
For his part, Putin made clear he was unhappy about the eastward expansion of the Western military alliance toward Russia. But he summed up his message to Bush and NATO this way: "Let's be friends, guys, and engage in an honest dialogue."
It was a striking change from Russia's once-angry threats to target missiles on Western capitals and Putin's unrelenting drive to clamp down on democracy, expand control over the government and the economy and quash independent news media.
Finished with talks in Bucharest, Bush flew to Zagreb to celebrate NATO's membership invitations to the Balkan countries of Croatia and Albania and an expected offer for Macedonia to join some day.
In his toast at dinner to welcome Bush, Croatian President Stipe Mesic, a staunch opponent of the war in Iraq, appeared to take a slight jab at the president by insisting that problems like terrorism, global warming and environmental destruction must be addressed jointly.
Bush, in his toast, congratulated Croatia on its NATO membership and praised it for sending troops to Afghanistan.
At a square in downtown Zagreb - far from the heavily guarded venues being used for the Bush visit - about 250 people held an anti-war protest, holding banners reading "USA NATO imperialism" and "The United States of Aggression."
On Saturday, Bush will go to Russia for a social dinner with Putin at the Black Sea resort of Sochi. On Sunday they will meet one last time before the Russian leader steps down on May 7.
Putin's handpicked successor, Dmitry Medvedev, will take part in some of the discussions. Putin is expected to continue to wield substantial power as Medvedev's prime minister.
It was seven years ago in June that Bush famously declared he had looked into Putin's soul at their first face-to-face meeting and found him to be honest, straightforward and trustworthy.
Relations grew stronger when Putin stood with the United States after the Sept. 11 terrorist attacks. But the era of cooperation quickly began to unravel.
Putin on Friday urged NATO leaders to listen to Russia's concerns, particularly its objections to NATO's plans to admit ex-Soviet republics Ukraine and Georgia. The two countries were not allowed to start on the path to membership this week, but leaders made clear they would be eventually.
"The emergence of the powerful military bloc at our borders will be seen as a direct threat to Russia's security," Putin said at a news conference after the meeting. "I heard them saying today that the expansion is not directed against Russia. But it's the potential, not intentions that matters."
"The efficiency of our cooperation will depend on whether NATO members take Russia's interests into account," he added. "We want to be heard, and we want see problems that divide us solved."
Still, he ruled out chances of a new Cold War and insisted that Moscow wants to be friends with NATO.
"None of the global players - Europe, the United States or Russia - is interested in returning to the past," Putin said. "And we have no ideological differences." He took note of Russia's agreement to facilitate transit of supplies for NATO forces in Afghanistan across Russian territory.
The White House said Bush repeated his frequent assurances that the Cold War is over and Russia is not the enemy.
Bush and Putin are expected to announce a "strategic framework" to guide relations beyond their time in office toward a less rocky future.
A major area of dispute is Bush's plan to build a missile defense shield in Europe to protect against what the United States sees as an emerging threat from Iran and other adversaries. Russia regards it as a move to undercut its nuclear deterrent.
Still, Putin praised Bush for trying to assuage Russia's concerns by offering to share information from the anti-missile system and to delay its activation until a nation like Iran tests a ballistic missile capable of reaching European cities.
"It's positive that they have heard our concerns," Putin said. He said the latest U.S. proposals show that "our U.S. partners are thinking about ensuring transparency and mutual trust."
Despite its strong rhetoric, Russia appears to accept that it is unlikely to stop the system. U.S. officials said missile defense will be part of the strategic framework but it remained an open question how far Putin was willing to go.
Putin criticized NATO nations for failing to ratify an amended version of the Cold War-era Conventional Forces in Europe treaty, which limits the deployment of aircraft, tanks and other heavy weapons on the continent. Russia last fall suspended its participation in the treaty, saying its old version became meaningless when nations in the former Soviet orbit joined NATO.
Putin said Russia wants to preserve the treaty, but that the West must ratify the new version signed in 1999. He dismissed NATO's claim that Russia needs to withdraw its troops from Georgia and Moldova, saying there is no link between the issues.
--- Associated Press Writers Vladimir Isachenkov, William J. Kole, Paul Ames and Claudia Kemmer contributed to this report from Bucharest.
