* Kremlin says US, Russia agree to step up cooperation
* Russia hopes for U.S. rethink on North Caucasus
* Washington wants to get reset in relations on track
Russian President Vladimir Putin may have been tempted to tell Barack Obama "I told you so" when U.S. officials blamed two ethnic Chechens for the Boston Marathon bombings.
He has long said the United States underestimates the security threat posed by Islamist militants in Russia's volatile North Caucasus, and has rejected criticism that Moscow's use of force in the region has been heavy-handed.
But the Kremlin leader has kept silent in public since U.S police killed Tamerlan Tsarnaev in a shootout and captured his younger brother Dzhokhar after a manhunt. Both are ethnic Chechens who had been living in the United States.
Instead, he and the U.S. president made positive statements about cooperation on counterterrorism in a phone conversation on Friday, suggesting both sides see an opportunity to improve strained relations between their countries.
"I hope the revelation of the bombers' Chechen ties will, if anything, open a window of opportunity to repair U.S.-Russia security cooperation," said Matthew Rojansky, deputy director of the Russia and Eurasia program at the Carnegie Endowment in Washington.
The Kremlin said in a brief statement after the phone call between Putin and Obama that the two presidents had agreed to step up cooperation on counterterrorism.
Putin's spokesman, Dmitry Peskov, sounded an upbeat note in a television interview, telling state-run Russia 24: "I think that there will be contacts between our intelligence services."
He gave no details. A U.S. law enforcement source told Reuters on Saturday that Russia had asked the FBI to investigate Tamerlan Tsarnaev in 2011.
It was not clear what Moscow had done to cooperate with Washington since Monday's bombings, but even small steps would be progress in a security relationship that has worsened in recent years.
Moscow and Washington have been at odds over the conflict in Syria and what the U.S. government sees as a clampdown on dissent since Putin's return to the Kremlin for a third term as president last May.
WASHINGTON WANTS RESET, KREMLIN WANTS RETHINK
The Kremlin appears to hope Washington will be forced into a rethink on the North Caucasus, even though the U.S. State Department said in its latest survey of human rights around the world that the rule of law was "particularly deficient" there.
Putin cemented his rise to power by crushing an independence bid by Chechnya in the second of two wars there, so is sensitive to any criticism of his handling of the Islamist insurgency that has now spread across the North Caucasus.
Robert Legvold, professor emeritus at Columbia University and a Russia expert, said the events in Boston would help increase U.S.-Russian cooperation because the sides would share intelligence and information about the suspects.
He underlined that Russia had been quick to rally behind Washington after the attacks on the United States on Sept. 11, 2001, and a similar deepening of security cooperation could take place now.
"I think in the end, the Russians under Putin want to keep the relationship as constructive as possible," Legvold said. "This episode is likely to be more positive than negative (for U.S.-Russian relations)."
Putin needs closer cooperation on security matters now because he wants to ensure the 2014 Winter Olympics pass off peacefully in the Black Sea resort of Sochi, which is close to the violence in the North Caucasus.
During a visit to Moscow by White House national security adviser Tom Donilon this week, the two sides avoided hostile public rhetoric in a sign they want to get the "reset" in relations, sought by Obama when he became president, on track.
The main obstacle to better ties in the past few months has been a row in which the United States passed legislation to punish Russians suspected of involvement in human rights abuses including the 2009 death of whistleblower Sergei Magnitsky in a Russian jail, and tit-for-tat moves adopted by Russia.
But political analysts say the former Cold War enemies have both shown that despite the dispute, they want to limit the damage to relations.