Gostkowski, who is coming off hip surgery that limited him to four games last season, served as the Patriots' starter since 2006, when he replaced All-Pro kicker Adam Vinatieri. Gostkowski is the franchise's all-time scoring leader with 1,775 career points.
Gostkowski, 36, became the Patriots' longest-tenured player after franchise quarterback Tom Brady signed with the Tampa Bay Buccaneers last week. According to Spotrac, the kicker was scheduled to count $4.8 million against the salary cap this season.
The Patriots, who have about $23 million in cap charges for players not on their roster, currently have no kickers on the team. Veteran Nick Folk, who filled in for the injured Gostkowski last season, remains an unrestricted free agent.
Gostkowski converted 374-of-428 field goal attempts (87.4 percent) with the Patriots. In the postseason, he was 31-for-36 on field goals.
With his departure, the longest-tenured player on the Patriots is special teams star Matthew Slater, who has been with the franchise for 12 seasons.