The Giants are expecting the punter to retire later this week and selected Matt Dodge, a punter from East Carolina, in the seventh round of Saturday's draft.
Feagles, 44, recently signed on to play his 23rd NFL season but last week told Tom Coughlin he might be unable to do so. Feagles will inform the Giants of his decision this week.
"He's 44 years old," Coughlin said. "He's having some physical issues and so he's decided to deal with them."
Feagles, who is the oldest active player in the league, has played in an NFL-record 352 consecutive games, which is 65 more than runner-up Brett Favre and the third-highest total in NFL history.
Coughlin said he would be surprised to hear that Feagles is planning on returning for 2010. That would leave Dodge and Australian rules punter Jy Bond to compete for the job in training camp.