The deal is for one year and is worth a maximum of $5.5 million, a source told ESPN NFL Insider Adam Schefter.
"Hakeem is a big, tough, and highly competitive wide receiver who knows what it takes to win," Colts general manager Ryan Grigson said in a statement issued by the team. "His big-play potential requires attention from the defense every time he's out on the field. He adds another highly talented playmaker to our offense."
Nicks was the New York Giants' first-round pick in 2009 and a significant contributor to their most recent Super Bowl title following the 2011 season. But leg injuries limited him significantly in 2012, and while he appeared to be healthy in 2013, he was not productive. Nicks failed to catch a single touchdown pass all year, although he caught 56 passes for 896 yards, and he seemed to draw the ire of the coaching staff for his role in the offense's struggles.
During the season, Nicks made repeated references to "my situation," meaning his contract situation, as an explanation for decisions he made about his workouts and preparation. He also switched agents in February, joining up with Jay Z's Roc Nation in advance of his free agency.
Nicks had recently said he was willing to take a one-year deal to prove he was healthy.
Nicks averaged 77.5 catches, 1,122 yards and nine touchdowns from 2010 to 2011 and just turned 26 in January. He has never played fewer than 13 games in a season, but he has never played a full 16, either.
With the Colts, he joins a receiving corps that includes perennial Pro Bowler Reggie Wayne and rising star T.Y. Hilton, and he'll play with one of the top young quarterbacks in Andrew Luck. Wayne will try to return this season from a torn ACL.
ESPN.com Giants reporter Dan Graziano and The Associated Press contributed to this report.