Bradham, 30, was informed of the Eagles' decision on Tuesday afternoon, the source said.
The move gives the Eagles an additional $4.5 million in cap room, upping their projected total to around $45 million in cap space as they enter the free-agency period.
Bradham was scheduled to make a base salary of $8 million in 2020.
Originally a fourth-round pick of the Buffalo Bills in 2012, the Florida State product signed with the Eagles in 2016 and became a key component of Jim Schwartz's defense, averaging 96 tackles over his first three seasons in Philadelphia.
He was slowed by an ankle injury this past season, however, totaling 61 tackles and an interception over 12 games.
In 115 games in his career, he has 619 tackles, 8.5 sacks, 3 interceptions and 5 forced fumbles.
The Eagles need to reload at linebacker. Kamu Grugier-Hill's contract is up, leaving Nate Gerry as the only linebacker remaining on the roster who played significant snaps on defense in 2019. T.J. Edwards, Alex Singleton and Duke Riley are the other remaining linebackers.