Feagles declined to say how much money he would earn, but he made $1 million last year in the final year of a five-year deal.
"I'm just very, very happy to have a chance to come back and play for the defending world champions," Feagles said in a telephone interview with The Associated Press. "It feels great."
Feagles spent 20 years in the NFL before making it to the Super Bowl this season, and he acknowledged he considered retirement briefly after the Giants 17-14 victory over the New England Patriots.
"Going through the season I proved to myself that I could continue to do this at a high level," Feagles said. "Certainly I considered riding off into the sunset after winning the Super Bowl. But I still have a fire to play football. Making my decision to stay easier was having my family living here. The wild card in the whole decision was the Giants, and they wanted me to come back. I can't say enough about the organization."
Feagles had a net average of 36 yards on 71 punts during the regular season. He landed 25 inside the opponents 20-yard line. In the Super Bowl, he averaged 39 yards on four punts, with only one being returned for 15 yards.
"Jeff is the best directional punter of all time," Giants general manager Jerry Reese said. "This guy can eliminate the opponent's punt returner. He is a pro's pro."
Feagles has not missed a game in 20 years, playing in an NFL-record 320 consecutive regular season contests. Jim Marshall, a defensive end who played for Minnesota and Cleveland from 1960-79, is second on the list with 282.
Feagles is the league's career leader for punts (1,585), yards (65,793) and punts inside the 20 (508). He has a career gross average of 41.5 yards and a net average of 35.7 yards.
The past nine days have been a blur for Feagles, with celebrations, parades and a stream of telephone calls and messages from friends and relatives.
"Each day it is starting to sink in a little more, with the reception of the people we see, all the things on TV and every time you hear 'world champions,"' Feagles said. "Every time you hear that we are the best in football, that is an incredible feeling, one that I had never felt. Every year 32 teams start the year looking for that. It was this year for us."
Feagles will be inducted into the University of Miami Hall of Fame on Wednesday. He was a member of the Hurricanes' undefeated national champions in 1987.
The Giants also are close to reaching a five-year, $7 million contract with kicker Lawrence Tynes, who kicked the game-winning field goal in overtime in the NFC title game against the Green Bay Packers.
"It probably will be in another day or two," said Gil Scott, Tynes' agent, said.
Tynes converted 23 of 27 field goal attempts this season and 40 of 42 extra points. He was 5 of 7 in the postseason, with the misses coming on game-winning attempts from 43 and 36 yards in the fourth quarter against the Packers on a frigid day in Green Bay when the temperatures reached minus 24. He converted from 47 yards in overtime to send the Giants to the Super Bowl.
