Johnson signed a minor league contract with the club in February. He will play with the Clippers and also for Double-A Akron, depending on the teams' schedules.
Johnson had offseason surgery on his right wrist and the Indians are eager to see if he can help them.
Johnson took batting practice and grounders at Progressive Field on Monday. Manager Manny Acta said Johnson needs "a lot of at-bats" in the minors before the Indians will consider him. Johnson, who is not on Cleveland's 40-man roster, played for Acta in Washington.
A .270 career hitter, Johnson played in just 24 games for New York last season.