It is a move that helps the salary-cap-strapped team create more space, but leaves a notable void at the position.
Gostkowski, who is coming off left hip surgery that shortened his 2019 season to four games, has been the team's full-time kicker since 2006, when he replaced Adam Vinatieri. He is the franchise's all-time leading scorer with 1,775 points.
Head coach Bill Belichick saluted Gostkowski in a statement Tuesday.
"I have had the privilege of coaching some of the best specialists in the NFL and Stephen Gostkowski is one of them," Belichick said. "Stephen had many highlights over his 14 years as a Patriot and one of the most impressive was the way it began, when he succeeded the most accomplished kicker in NFL history. For some, this may have been too difficult a challenge. But from his rookie year, Stephen exuded a maturity and confidence that demonstrated he was up to the task. He made several crucial kicks his rookie year and established a top level of performance and consistency for years and years to come.
"Stephen is a great teammate who made outstanding contributions over a decade of championship football."
Gostkowski, 36, was the Patriots' longest-tenured player once quarterback Tom Brady signed with the Tampa Bay Buccaneers last week. Gostkowski was scheduled to count $4.8 million against the salary cap in 2020.
The Patriots, who have nearly $23 million in cap charges for players not on their roster (which includes $13.5 million for Brady), don't have another kicker on their roster. Veteran Nick Folk finished last season in that role and remains an unrestricted free agent.
Gostkowski was 374-for-428 on field goals (87.4%) with the Patriots, and 653-for-664 on point-after attempts (98%). That didn't include the playoffs, where he was 31-for-36 on field goals and 79-for-83 on PATs. His strong right leg was also valuable to the team on kickoffs.
Only Tom Brady (41), Adam Vinatieri (32) and Jerry Rice (29) have played in more career playoff games than the 28 of Gostkowski, whose 205 postseason points are second all-time to the 238 by Vinatieri.
With his departure, the longest-tenured Patriots player is special-teamer Matthew Slater, who has been with New England for 12 seasons.