Tamerlan, 29, died last Thursday in an exchange of fire with police in a Boston neighbourhood.
He is suspected to be the mastermind of the blasts that killed three people and injured 264 others on April 15.
His accomplice and younger brother, Dzhokhar, has been arrested and is being treated in a Boston city hospital.
"Tamerlan Tsarnaev did ping in our systems when he travelled. But we also know that the FBI did a thorough investigation in 2011 and did not find any terror activity, foreign or domestic," White House Press Secretary Jay Carney told reporters at a news conference.
He was responding when asked why the FBI did not follow the Boston bombing suspect when it was alerted by the Russians about his radicalisation.
"You know, as the President made clear when he spoke before you on Friday night, he wants every agency involved in this to do a broad investigation into what happened, what we knew, what inspired and motivated these two individuals, and the steps that they took that led to the terrorist attacks in Boston a week ago Monday,” he said.
"That process needs to take place, and it's being undertaken now in an investigation led by the FBI and a prosecution, obviously, led by the justice department," he said.
The FBI, he said, took action in 2011 in response to the information provided to them by the Russian government, and the conclusions that they reached then and the nature of the investigation they conducted then.
"Beyond that, I think we need to let the investigation unfold and make assessments when we know all the facts," Carney said.
The US Embassy in Russia, he said, has sent a team down to Dagestan to interview Tsarnaevs' parents.