Only a week after losing an All-Star closer, the Twins might be about a week away from getting one back.

Glen Perkins, who hasn’t pitched in a major league game in 16 months after surgery to reattach the labrum in his left shoulder, believes a rehab stint with Class AA Chattanooga next week could be the final step of his long journey back to the Twins.

“If I can get through this next weekend, there’s no reason I shouldn’t be ready to play here,” Perkins said in the Twins clubhouse Sunday morning, after presiding over his annual Fifteen’s 5K charity race. “It’s been frustrating, but I’m almost there.”

Perkins, who has appeared in five minor league games since July 22, will throw 30-35 pitches in the bullpen for Twins coaches Monday, then depart Wednesday for Birmingham, Ala., where he will join the Lookouts. He’s scheduled to pitch at least an inning in Thursday’s game, then try to go back-to-back in games Sunday and Monday — what he called “the final test.”

“Since I had a cortisone shot [three weeks] ago, it’s been steady progress,” said Perkins, who underwent shoulder surgery in June 2016. “It’s been really good. I haven’t had a bad day. … It’s been encouraging. I just keep my fingers crossed it keeps up this way.”

So do the Twins, who traded away All-Star closer Brandon Kintzler, at the trade deadline last week. Though Perkins might not be in line for that job anymore, when he demonstrates he is ready, he will be activated, Derek Falvey confirmed.

“He’s bounced back from his outings much better of late, in a way that gives you hope that he feels good and he’s about ready,” said Falvey, the Twins chief baseball officer. “That’s been the goal all along. It hasn’t been as linear a path as he expected, but it’s a serious surgery that takes a long time. This week will be good — a couple more outings, and a back-to-back, just to make sure.”

Could Perkins, an All-Star from 2013 to ’15, be back in the bullpen 10 days from now?

“It’s certainly possible,” Falvey said, “but let’s let this week play out.”

Illness sidelines Dozier

Brian Dozier hit a big home run to help the Twins rally to beat Texas 6-5 on Sunday, but he wasn’t around at the finish. The second baseman hit a 2-0 fastball into the left-field seats in the second inning, cutting the Twins’ three-run deficit to 5-4 with a two-run shot.

He left after the fifth inning, however, after starting to feel sick.

“He was lightheaded, dizzy. He felt a little faint, to be honest,” manager Paul Molitor said. “I didn’t want to take any chances. He’s responded, he’s feeling better. Hopefully, there’s no repercussions tomorrow.”

In Dozier’s absence, Ehire Adrianza made perhaps the game’s most critical defensive play. With Elvis Andrus on third base in the seventh inning after left fielder Eddie Rosario lost his catchable fly ball in the sun, Adrian Beltre lined a two-out pitch from Trevor Hildenberger up the middle. Adrianza made a reaching, running grab of the liner in short center field, preserving the Twins’ one-run lead.

Double-checking Sano

Third baseman Miguel Sano will undergo a new series of tests — an X-ray, CT scan and magnetic resonance imaging exam — on his sore left hand Monday, just to make sure that swelling didn’t mask anything more serious than a bruise.

Sano has been out of the lineup since being hit by a pitch on his hand Friday, but the disabled list is not being contemplated yet, Molitor said. Still, Sano was unavailable for Sunday’s game.