Francona said Bourn might suit up for Triple-A Columbus, which is scheduled to open its season Thursday.
"We're going to talk to Bourny when we get back," Francona said. "I just know, when look at the reports and I see him doing well, I feel good, because that means he'll be back. Whether he's back on the day or a day late, that's not the end of the world."
Bourn is eligible to come off the DL on Saturday.
He injured his left hamstring while running the bases in a Cactus League game against the Giants on March 16. Francona said the center fielder went 2-for-3 with a pair of doubles in a Minor League game Sunday and he was able to chase down a fly ball in the outfield.
"From all the reports," Francona said, "he did really well."
Bourn, 31, signed a four-year contract worth $48 million with the Indians prior to last season, when he hit .263 with six home runs, 21 doubles, six triples, 50 RBIs, 23 stolen bases and 75 runs in 130 games. Bourn underwent surgery on his left hamstring on Oct. 15, but the team has indicated that the current injury is unrelated to that issue.
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.