Gostkowski, the Patriots' all-time leader in points and field goals, has been placed on injured reserve. He is expected to be ready for the 2020 season, the source told Schefter.
"He's meant a lot to this team, and you see a lot of that in the record books," veteran safety Devin McCourty said. "... But I always speak on how much he means to the locker room. So he'll definitely be missed throughout the year, having his presence in there."
Gostkowski, 35, who has been the Patriots' kicker since 2006, missed four extra- point attempts during the team's 4-0 start. He was 7-for-8 on field-goal attempts. He also handled kickoffs, and he was tied for second in the NFL with 21 touchbacks. Rookie punter Jake Bailey, who has experience with kickoffs, could take over those duties.
Other kickers who auditioned for the job were Elliot Fry, Blair Walsh and Matthew Wright.
The Patriots could wait another day to make a final decision, which would give them time to work out other free agents who weren't able to make it into town Wednesday.
Forbath, who has kicked for the Redskins, Saints, Vikings and Jaguars, is 120-for-140 on field-goals attempts in his career.
Nugent is 253-for-311 on field goals in his career. He has been with the Jets, Buccaneers, Cardinals, Bengals, Cowboys, Bears and Raiders.
This will mark the first time since 2010 that the Patriots have had a kicker other than Gostkowski, who landed on injured reserve that season due to a thigh injury and the team signed veteran Shayne Graham to fill the void.
"Whoever the next guy is that comes in here, he doesn't need to try to be Steve or try to be Adam Vinatieri; he just needs to be himself," special-teams captain Matthew Slater said. "He'll be here for a reason and that reason will be because Coach Belichick thinks he's going to give us the best chance to help this team win games. We'll support him. We'll do what we can to help him adjust."