MESA, Ariz. — A potentially major figure for the A’s this season will be almost invisible this spring: starting pitcher Henderson Alvarez, signed this winter after being non-tendered by the Marlins, is rehabbing after 2015 shoulder surgery and won’t appear in any Cactus League games.

Once Alvarez is healthy, though, he might play a big role in Oakland’s fortunes. He was an All-Star in 2014 with the Marlins, putting up a 2.65 ERA, and he threw a no-hitter on the final day of the 2013 season.

He’s only 25, and he could be the A’s second-best starter when he comes off the DL.

Alvarez indicated that might be sooner than forecast; the A’s have said a June return is most likely, but Alvarez said Saturday that he believes that he could be throwing in games in April. He’ll need a number of starts in the minors, of course; still, that means a potential May debut for Alvarez with the A’s.

“If you watch him play catch, it looks like there’s nothing wrong with him,” manager Bob Melvin said. “That was a great pickup, he’s a guy who’s pitched one or two on staffs before. We want to make sure he’s healthy, but we’re really excited about having him.”

Alvarez threw at a distance of 135 feet Friday, and he figures he’ll be throwing off the mound in March. He’s looking forward to joining Sonny Gray in the A’s rotation. “Sonny is great,” Alvarez said. “That’s nasty, good stuff.”

Injury report: Right-hander Jesse Hahn, shut down the second half of last season with a forearm strain, said he’s thrown off the mound five times and everything is going well. “The biggest thing is the mental side of it,” Hahn said. “It’s a relief.” ... Right-hander Jarrod Parker, who is coming back from his second Tommy John surgery and a fractured elbow, said he has thrown three bullpen sessions and he’s scheduled for another Monday. The plan is still to stretch him out to start, Melvin said.

Briefly: Melvin said outfielder Coco Crisp will get plenty of playing time even with the acquisition of outfielder Khris Davis from Milwaukee this month. “It’s all about health for Coco. If he’s playing well, we’re going to find at-bats for him,” Melvin said. … Mark Canha also seemingly is squeezed out a bit with the Davis and Yonder Alonso pickups, but Melvin said that there should be plenty of opportunity for Canha, too, and he emphasized that anyone who is performing well will be in the lineup. … Gray said he’s about 5 pounds heavier this year, at 185. “He feels like he needs to be a little bit heavier, a little bit stronger,” Melvin said. “Toward the end of the season, he’s been a little beat up a couple of years.” … The A’s are likely to have Gray throw bullpen sessions the first week or so of Cactus League play rather than pitch in games. ... Melvin plans to get top pitching prospect Sean Manaea a start on a split-squad day and believes Manaea could help the A’s this season.

Susan Slusser is a San Francisco Chronicle staff writer.