Morneau, who is on the disabled list retroactive to May 1 with a sore left wrist, had a cortisone shot in his wrist. Dr. Thomas Graham, an orthopedist with the Cleveland Clinic, gave him the shot on Tuesday.
"He's supposed to be [swinging] tomorrow or if he's not feeling OK, we could back it up a day," Gardenhire said. "The pace, I don't know if we have a formula here. We're going to let him tell us when he'll be ready. He's going to grab a bat in his hand and once he feels good enough, he'll start taking some swings. And when that happens, I'll tell you."
There's still no official timetable for Morneau's return, but he could make his return during the club's eight-game road trip that starts Wednesday in Detroit.
"I don't think he's been out long enough to say he'll need a rehab assignment," Gardenhire said. "If he starts swinging and he feels good, I'd rather him start up here than somewhere else."