Indianapolis waived defensive tackle Jordan Thompson in a corresponding transaction. Kelly, 25, was the final pick of the 2017 NFL Draft out of Mississippi.
The 6-foot-2, 228-pound quarterback tried out for the Colts in early May at rookie minicamp. Kelly took just one snap during the 2018 season. He kneeled for a one-yard loss during the Denver Broncos' Week 6 loss to the Los Angeles Rams.
Kelly entered the league with the Broncos, but did not take a snap during the 2017 season. He was cut by the Broncos in October, following an arrest.
The Colts also have Phillip Walker and Jacoby Brissett on the quarterback depth chart, behind starter Andrew Luck. Walker was signed in 2017 as an undrafted free agent out of Temple. Brissett joined the Colts in a 2017 trade from the New England Patriots. The three-year veteran is entering the final year of his rookie contract.
Kelly completed 62.5 percent of his throws for 10,763 yards, 97 touchdowns and 29 interceptions during his collegiate career at Mississippi, East Mississippi Junior College and Clemson.