Bowman will have surgery to repair the ACL, but the MCL is expected to heal on its own, the source said, adding that Bowman is expected to be ready for the start of the 2014 season.
Bowman was injured during the fourth quarter of San Francisco's 23-17 loss to the Seahawks in Seattle when two players fell onto his planted left leg as he made a play on a goal-line stand.
He appeared to have a fumble recovery on the play, but officials ruled Seattle running back Marshawn Lynch ended up with the ball.
A candidate for the NFL Defensive Player of the Year Award, the inside linebacker was taken off the field on a cart.
Bowman, a four-year veteran, led the Niners in tackles this season with 145.
ESPN.com San Francisco 49ers reporter Bill Williamson contributed to this report.