Nugent, 37, was one of seven kickers the Patriots worked out over the past two days.
He is 253-of-311 on field goal attempts in his career, having kicked for the New York Jets, Tampa Bay Buccaneers, Arizona Cardinals, Cincinnati Bengals, Dallas Cowboys, Chicago Bears and Oakland Raiders.
Nugent, who has appeared in five playoff games, was the most experienced kicker to work out with the Patriots.
He beat out Kai Forbath, Blair Walsh, Elliot Fry, Matthew Wright, Younghoe Koo and Josh Gable for the job that came open when Gostkowski (left hip) was placed on injured reserve Wednesday. Gostkowski is the Patriots' all-time leader in points and field goals.
Koo, who last kicked as a rookie for the Chargers in 2017, was added to the Patriots' practice squad as insurance, a source told Schefter. In four career games with the Chargers, Koo was 3-for-6 on field goal attempts while converting all nine of his extra point attempts.
The Patriots likely will have rookie punter Jake Bailey handle kickoffs.
Gostkowski, 35, who has been the Patriots' kicker since 2006, will have season-ending surgery and is expected to be ready for the 2020 season, a source told Schefter on Wednesday. He missed four extra point attempts and was 7-for-8 on field goal attempts during the team's 4-0 start.
This will mark the first time since 2010 that the Patriots have had a kicker other than Gostkowski, who landed on injured reserve thanks to a thigh injury that season, and the team signed veteran Shayne Graham to fill the void.