He may miss only four regular-season games because of the injury, a tight left hamstring.
Bourn had surgery on the hamstring in October. He felt discomfort on March 16 while running from first to third against San Francisco, and has been undergoing treatment and started strength exercises.
"He is getting close to being game ready,'' general manager Chris Antonetti said Monday. "We want to be able to build up his innings, and his progressions so when he's back he will be able to impact the team.''
Outfielder Nyjer Morgan was told by Antonetti and manager Terry Francona that he will be on the Indians' Opening Day roster.
The Indians did not want to rush Bourn back in light of the surgery.
"We involved Michael in the decision,'' Francona said. "He is very intelligent and conscientious. As we walked through it with him it just seemed to make sense to start him on the disabled list. All of our goals combined is to have him back being Bournie, not half of Bournie.
"If we asked him to he'd play, no doubt about that. He'd play with one leg hanging off. We want him to build up properly so he can really help us,'' he said.
Morgan was signed to a minor league contract by the Indians on Jan. 16. He spent last season in Japan with Yokohoma.
"It's a very exciting day for me. It's unfortunate what is happening with Michael,'' Morgan said. "I'm going to be a great teammate. Whenever he comes back, I will slide to the back or whatever they want me to do. I wanted to come out here and prove to everybody that I'm a great player. I worked hard this offseason. I feel mentally and physically prepared. The hard work is definitely not going to stop.''
The Indians made no promises to Morgan when they signed him.
"We told Nyjer that we would give him at-bats and give him a chance,'' Antonetti said. "He's had a really good camp. Offensively and defensively, the way he's gone about his business day to day.''
Cleveland told third baseman Lonnie Chisenhall he will join the team for its opener.
The Indians told catcher Luke Carlin that he will not start with the major league club on Opening Day but will remain in camp the last week of spring training.
Right-hander Aaron Harang was given his unconditional release.