Jagger had vowed not to use Hells Angel members as bouncers following the death in December 1969 of an 18-year-old fan at a notorious free performance at Altamont Speedway in Northern California.
In return, gang members hatched a plan to kill Jagger at his holiday home in Long Island, New York, the BBC claimed.
"The Hells Angels were so angered by Jagger's treatment of them that they decided to kill him," Tom Mangold, the presenter of the program, was quoted as telling Britain's Sunday Telegraph newspaper.
He said the plan was disclosed during an interview with Mark Young, a former FBI officer, for the BBC's "The FBI at 100" documentary.
Mangold said the men tried to reach Jagger by sea. "The boat was hit by a storm and all of the men were thrown overboard," he was quoted as saying. They all survived but made no other attempt on his life, Mangold said.
It was not clear whether Jagger was ever informed of the alleged plot against him.
LD Communications, Jagger's publicists in Britain, did not immediately return calls requesting comment.
The Hells Angels have always denied any connection with the Altamont Speedway killing.