The Bengals also brought back Wallace Gilberry, agreeing with the defensive end on a three-year deal worth $6.75 million, a league source told ESPN NFL Insider Adam Schefter.
A nine-year veteran, Nugent has been with the Bengals for three seasons. He injured his right calf in December but Josh Brown filled the void during Cincinnati's playoff run, converting 11 of 12 field goals (91.7 percent), with his only miss coming on a desperation 56-yarder in Pittsburgh.
Nugent, who received the Bengals' franchise tag last offseason after setting team records with 132 points and 33 field goals in 2011, turned 31 last week while Brown turns 34 next month. He tied a Cincinnati record with a 55-yard field goal against Oakland last season.
Nugent has made 83.8 percent of his field goals as a Bengal, the second-best career percentage in club history.
"It's great to be able to come back to a place where you love going to work every day," Nugent told the team's website. "That obviously means a lot and I really think we've got a good thing going with the specialists. We've been together a long time and that not only helps us, but it helps the team, too."
Gilberry, 28, recorded 6.5 sacks last season, his first with the Bengals. The five-year veteran has 20.5 sacks in 67 career games with Cincinnati and Kansas City.
Information from ESPN NFL Insider Adam Schefter, ESPN.com AFC North blogger Jamison Hensley and The Associated Press was used in this report.