The five-year contract includes $4.7 million in guarantees and a maximum value of $11.165 million.
Agent Ian Greengross confirmed the deal, which Addai is expected to sign Sunday. The Colts are scheduled to report for training camp Sunday and begin practice Monday.
Addai, who split time at LSU during his college career, had one of the fastest 40-yard dash times at the NFL scouting combine in February. The Colts believe he can be a breakaway threat in their offense.
Addai was the No. 30 overall pick and is expected to compete with Dominic Rhodes for the starting job. James, the Colts' all-time leading rusher, left the team for Arizona as a free agent this offseason.