Terms were not disclosed, but a source told ESPN's Adam Schefter that Hunt agreed to a two-year, $9 million deal that could be worth up to $10 million with incentives. Hunt was scheduled to become an unrestricted free agent next week.
Hunt, 31, told WTTV-TV in Indianapolis that it was a "no-brainer" to re-sign with the Colts.
"If that kind of offer comes up before free agency, they clearly were trying to keep me here,'' Hunt said. "The offer that was put in place sounded great.
"I didn't want to leave this kind of locker room behind and where it's headed.''
Hunt went from being predominantly a special-teams player and starting only five games during the first five years of his career to starting all 15 games he played with the Colts last season.
His versatility of playing on the edge and on the interior of the defensive line helped keep him on the field last season. That also played a key factor with him having a career-high five sacks to go with 30 tackles.
All 20 of Hunt's career starts came during the 2017 and 2018 seasons with the Colts. The former second-round pick spent his first four seasons in the NFL with Cincinnati.
ESPN's Mike Wells contributed to this report.