Celek was the longest-tenured player on the Eagles, who cleared $4 million from their salary cap with the move.
The 33-year-old has appeared in all but one game with the team over his 11 NFL seasons. His role has been altered significantly with the emergence of Zach Ertz, with Celek finishing with a career-low total in catches (13) and receiving yards (130) last season.
"Brent Celek defines what it means to be a Philadelphia Eagle. His dedication to his profession and this organization is unmatched and he will go down as one of the best tight ends in franchise history," the team said in a statement. "Brent embodied the City of Philadelphia's temperament and character with his toughness and grit. He has been a huge part of everything we have been building over the last decade and it is only fitting that he was able to help us win our first Super Bowl last season.
"Unfortunately, in this business we are forced to make difficult decisions, especially this time of the year. This one is as tough as they come, but in our eyes, Brent will always be an Eagle."
Celek's departure comes on the heels of reports surfacing that fellow tight end Trey Burton will sign a four-year contract with the Chicago Bears. Burton etched his name into Philadelphia lore last month for his part in the "Philly Special" play during Super Bowl LII.
Celek set career highs in 2009 with 76 receptions for 971 yards and eight touchdowns. For his career, Celek has 398 catches for 4,998 yards and 31 scores.
He was a fifth-round draft pick of the Eagles out of Cincinnati in 2007.