There is a team option on the third season.
The Oklahoman newspaper earlier reported the deal.
Morrow, 28, is coming off a productive season with the New Orleans Pelicans where he averaged 8.4 points and shot 45.1 percent from 3-point range, his best year since his 2011-12 season with the Nets.
The Thunder have desperately been trying to fill a need on the wing with a shooter, and have found a good one in Morrow, who is a career 42.8 percent shooter from deep. Former starter Thabo Sefolosha signed with the Atlanta Hawks, opening more of a hole in the backcourt. Morrow could potentially slot in as the starter, or provide depth off the bench. Morrow will provide spacing for Kevin Durant and Russell Westbrook, who are two of the league's best attackers with the ball. The Thunder ranked 14th in the league in 3-pointers made per game, but they were unbalanced in that regard, with Durant providing the bulk of them.
ESPN.com's Jeff Goodman contributed to this report.