Francona said it will probably take a little longer.
"The best-case scenario is he misses six days, four games," Francona said. "I wouldn't be surprised that, even if he's ready to play, maybe it's a few more days just so he can be the player he wants to be. He's willing to go out there, which we appreciate.
"But at the beginning of the season, with so many games left to play and having that speed component being so important, we need to let him feel good about himself."
Bourn, who tweaked his hamstring during a Cactus League game against the Giants on March 16, remained in Arizona and recently resumed running the bases.
Veteran Jason Giambi -- in line to serve as a pinch-hitter and part-time designated hitter for Cleveland this season -- continues to progress in his comeback from a fractured rib in his right side. Giambi, who broke the rib on a hit-by-pitch on March 7, hit in the batting cage on Friday and continued to hit off a tee and do soft-toss drills.
Giambi is also in line to be placed on the 15-day DL on Sunday and is hoping to be able to return when eligible on April 5. Giambi needs to take regular batting practice (currently scheduled for Monday in Oakland), and he said the Indians want to also simulate a pinch-hitting scenario before activating him.
"Everything feels great, so we'll just keep going," Giambi said. "I know they want to still do one time where I hit and then have to act like I have to pinch-hit. [I'll] sit kind of for a while and try to get it going, because that's where I'm beneficial for the ballclub as well."
Francona has been encouraged by Giambi's progress.
"He's doing really well," said the manager. "Everything he's doing right now, he's not feeling any pain. That's really good. There's got to be a little bit of a progression, but G's not playing every day, so we don't have to run him out there every day. But we do have to do a progression where it's fair to him, so he can help us."
Jordan Bastian is a reporter for MLB.com. Read his blog, Major League Bastian, and follow him on Twitter @MLBastian. This story was not subject to the approval of Major League Baseball or its clubs.