SCOTTSDALE, Ariz. — Tony Watson, who would be in the mix for the closer’s job — if indeed the Giants have a closer’s job — is anticipating his shoulder tightness won’t prevent him from being ready for the season opener.

“It’s nothing I’m concerned about,” said Watson, who on Monday threw his second bullpen session in three days. “Just need to be smart about it.”

Watson, 34, is behind schedule after it was detected early in camp that his shoulder strength wasn’t where it should be. He took a break from throwing and worked himself back. He’s now throwing his first bullpen sessions. It’s not known when he’ll appear in a Cactus League game.

“The most important thing is to be ready for when the season starts, Opening Day,” Watson said.

Asked if that would be no problem, Watson said, “Right now, the calendar’s working in our favor, but we can’t have any setbacks. Everything’s got to bounce out. I’m trying not to look too far ahead. The games are going by. Hopefully, we keep progressing.”

Watson threw fastballs and changeups Monday — “basically like the first ’pen when you get here. It’s just three weeks later than normal” — and manager Gabe Kapler said the left-hander will have another bullpen session before facing hitters.

In a perfect world, Watson would face hitters twice and then begin making Cactus League appearances. He also has been throwing long toss.

“Everybody battles certain things throughout the year,” Watson said. “Fortunately, this one was early in camp. It’s never good timing, but this is about the best timing it could be.”

Watson has 30 career saves to his name, though none in the past two years as a Giant. No one has been named to replace All-Star closer Will Smith, who signed with the Braves, and it’s possible the Giants will open the season without one person assuming the role.

Kapler managed in Philadelphia the past two years, and 14 of his pitchers combined to earn 80 saves, though two of them, Hector Neris (39) and Seranthony Dominguez (16), earned the lion’s share.

Kapler would prefer flexibility from his relievers rather than have them focus on defined roles.

John Shea covers the Giants for the San Francisco Chronicle. Email: jshea@sfchronicle.com Twitter: @JohnSheaHey