Damon left Sunday's 4-2 loss to Minnesota with a bruise on the tip of his left ring finger. He hurt himself while trying to bunt in the third inning.
Damon had a splint on the finger Monday, and hopes to be able to grip a bat in the next day or two. He hasn't ruled out playing later this week.
"We definitely have to wait for the swelling to go down," Damon said. "I think we're doing everything necessary to do so. We'll see how it feels. I'll be back in there soon."
Also, Rays All-Star third baseman Evan Longoria, on the 15-day disabled list since April 3 because of a strained left oblique, could start a minor league rehab assignment next week and rejoin the team by the end of the month.
"It's possible," Tampa Bay manager Joe Maddon said.
Reliever J.P. Howell, coming back from left shoulder surgery, is scheduled to throw his third batting practice session Tuesday and could pitch in the first of seven or eight minor league games late this week.