By Steve Gutterman

MOSCOW (Reuters) - U.S. lawmakers on a mission to Russia said on Sunday they had found no evidence that an American intelligence error enabled the Boston bombings, but that closer cooperation between Washington and Moscow might have helped to thwart the attack.

U.S. investigators suspect two brothers who emigrated from Russia, one since shot dead by police, staged the attack at the Boston Marathon on April 15 that killed three people and wounded 264 others.

Two congressmen on the fact-finding visit said the countries - former Cold War foes now at odds over issues from Syria to President Vladimir Putin's treatment of opponents - had to work together better against a shared threat from Islamist militants.

"Radical Islam is at our throat in the United States, and it is at the throat of the Russian people," said Republican Representative Dana Rohrabacher, who led a group of six U.S. lawmakers on the weeklong visit to Russia.

President Barack Obama's administration and U.S. intelligence have faced scrutiny over claims they failed to see the danger from the suspects, Tamerlan and Dzhokhar Tsarnaev, ethnic Chechens who emigrated with their parents a decade ago.

"We've been asked a number of times, do we believe that the Boston Marathon massacre could have been thwarted - could it have been prevented? And the answer is, there's nothing specific that could have been done that we can point to that, had it been done differently, would have prevented this," Rohrabacher said.

"But we can say that had we had a much higher level of cooperation all along, so that the whole situation would have been different, I believe that would have been one of the type of things we could have thwarted," he told a news conference at the U.S. Embassy in Moscow.

The U.S. lawmakers met Federal Security Service (FSB) officials and visited the North Caucasus town of Beslan, scene of a deadly 2004 school siege some Russians call their country's equivalent of the September 11, 2001 attacks on the United States.

"The Cold War is over now, so we have to make friends with the Russians and recognize there is a mutual threat to both of us," Rohrabacher told the news conference. It was attended by U.S. action film actor Steven Seagal, a friend of Putin's who helped arrange the representatives' meetings in Russia.

U.S. officials have said Russian security services asked the FBI about Tamerlan Tsarnaev in early 2011 out of concern he had embraced radical Islam and would travel to Russia to join insurgents.

FBI agents interviewed him in Massachusetts in 2011 but said they found no serious reason for alarm. U.S. officials say Russia's FSB security services later failed to respond to the FBI's requests for more information about him.

SECURITY AND RIGHTS

Reading from notes from a briefing with FSB officials, Republican Representative Steve King said they indicated the FSB had told the FBI that Tamerlan was "very close to radical Islam and very religious".

"I suspect that he was raised to do what he did," King said of Tamerlan Tsarnaev, 26, who died in an April 19 shootout with police. Dzhokhar, 19, is in a Massachusetts prison hospital awaiting trial on charges that can carry the death penalty.

Tamerlan Tsarnaev travelled to Russia early in 2012 and spent six months in Dagestan, a North Caucasus province that is now at the centre of the Islamist insurgency rooted in two post-Soviet separatist wars in neighboring Chechnya.

The FBI did not tell the FSB that Tamerlan Tsarnaev had returned to Russia, the congressmen said. Representative Steve Cohen, a Tennessee Democrat, said it appeared the FSB had been unaware that he was in Dagestan.

Rohrabacher, asked whether U.S. authorities reacted appropriately to the information conveyed by Russia, said: "I think that given the circumstances and the level of cooperation, I would say they did - but I would also say that the level of cooperation was unacceptable."

He said some in the U.S. intelligence community are "still playing Cold War games" and that Putin is overly demonized.

The Kremlin has called for closer intelligence cooperation after the Boston bombings and high-level meetings have been held, but Russia's expulsion of an alleged U.S. spy recruiter last month underscored persistent tension.

U.S. officials have said they consider counterterrorism information from the North Caucasus suspect because Russian "watch lists" often include dissidents and rights activists mixed together with militants.

Cohen said the United States must not give Russia a free ride on human rights, an issue that has clouded relations since Putin started his third Kremlin term last year.

He disagreed with Rohrabacher and King over the jailing of members of punk band Pussy Riot for a protest in a church, calling the trial unfair and the two-year sentences excessive.

"We have a role in the world and we need to continue to observe that role as a place of ... due process and justice and fairness," Cohen said of the United States.

(Editing by David Stamp and Christopher Wilson)