The deal is for three years, a league source told ESPN's Field Yates.
"John possesses a strong balance between his pass rushing ability and run defense," Colts general manager Chris Ballard said in a statement. "He's a physical linebacker who has matured on one of the top defensive units in the league over the past three seasons. John provides a significant upgrade to our defense and we're excited to welcome him to Indianapolis."
Simon played the past three seasons with the Houston Texans. With Jadeveon Clowney's permanent move to defensive end after J.J. Watt's season-ending back surgery in late September, Simon played a more consistent role at outside linebacker.
He missed five games last season because of chest injuries, returning for the Texans' Week 16 matchup against the Cincinnati Bengals and sitting out Week 17 after the team already had clinched the AFC South title.
"John is a smart, physical player who will make an impact immediately on our team," Colts coach Chuck Pagano said. "We witnessed his success in Houston and now I look forward to seeing him wear the horseshoe. He will be a great addition."
Simon had 51 tackles in 11 games last season after posting 53 stops in 2015, during which he played in all 16 games.
The 26-year-old said he and the Texans did everything they could every day to get him back on the field faster while he was dealing with the injuries, but "it just wasn't healing too fast."
ESPN's Sarah Barshop contributed to this report.