The Russians only turned over the information to their US counterparts in recent days, according to the report.
A US official with knowledge of the investigation who confirmed the intercept to the US television network declined to confirm who was on the other line.
Asked about the report as he attended the White House Correspondents' Dinner, Attorney General Eric Holder declined to comment, saying it was an "ongoing matter."
The conversation raises the possibility that red flags were missed that could have alerted US officials to the threat posed by the Tsarnaev brothers, accused of killing three people and wounding 264 during twin blasts at the Boston Marathon this month.
Both the CIA and the FBI flagged deceased Boston bombing suspect Tamerlan Tsarnaev over possible terror ties after Russian officials contacted the US agencies in 2011.
Reports said Russian authorities had also alerted their US counterparts about concerns that his mother, Zubeidat Tsarnaeva, was a religious extremist, and that she was added along with her older son to a terror watchlist.
At the time, the FBI said Russia did not respond to US requests for additional information, and US investigators saw no reason to pursue the investigation further.
US authorities, meanwhile, have combed through a landfill for possible clues about the attack.
The surviving bombing suspect, Tsarnaev's 19-year-old brother Dzhokhar, is being held at a federal prison medical centre where he was transferred early Friday after spending several days in the same hospital as some of his alleged victims in Boston.