NASHVILLE — The A’s turned to a member of the world champion Royals to bolster their bullpen, signing right-hander Ryan Madson to a three-year deal Sunday.

And to boost its rotation, Oakland might reunite with one of its former All-Stars. There have been indications since late last season that the A’s have interest in re-signing Scott Kazmir, who was traded to Houston in July, and Sunday, Kazmir told The Chronicle via text that Oakland is one of the teams he’s talking to.

Numerous other clubs are pursuing the left-hander, including teams with bigger payrolls, but Kazmir enjoyed his time in Oakland and has many close friends there, including starter Sonny Gray and reliever Sean Doolittle.

“Oh, man, you can’t put the cart before the horse, but that gets me excited,” Doolittle said by phone. “When we played Houston in September, we were all giving him our best sales pitch, laying it on thick. We would all love to see him back at the Coliseum, not just from a baseball standpoint but from a leadership standpoint.”

Whether the A’s can afford Kazmir, given the deals that starters have received the past several days, remains to be seen. General manager David Forst has said that arms are a priority and that payroll is not an issue, but Kazmir, 31, is likely to be in the same neighborhood as new Giants starter Jeff Samardzija, at $17 million to $18 million per season.

Madson’s deal is worth $22 million, and, according to ESPN’s Jerry Crasnick, Madson, 35, can earn up to an additional $1.25 million per season based on games finished.

There’s little doubt that the A’s had interest in Madson because he can close if needed. Doolittle, Oakland’s All-Star closer, missed most of 2015 with a shoulder injury. He returned at the end of last season fully healthy, but the team would like to be able to lighten his load when necessary.

“I get it,” Doolittle said. “If we’ve learned anything from watching the postseason the past couple of years, it’s that you can’t have too many assets in the bullpen, and if you have multiple guys who can close, that’s not a bad thing.”

Doolittle, who became a Phillies fan as a child when his family moved to New Jersey, is a big fan of Madson.

“Of course!” he said. “He was a big part of those really good Phillies teams, part of that core of young players that all came up together.

“Obviously, that stuff he did with the Phillies carries a lot of weight, but the thing I have the most respect for is his comeback story. He missed three years and had all those hurdles to overcome, and I can kind of relate to that. Then to see his performance last year after missing three years — that was incredible. It speaks to his commitment to the game and mental toughness.”

Madson had Tommy John surgery in 2012 and, after failed comeback attempts, he briefly retired in 2014. Then he signed a minor-league deal with the Royals, made the club and put up a 2.13 ERA.

He appeared in three games in the World Series and didn’t allow a run while striking out five in three innings. His WAR of 1.7 put him in the top five of available relievers, and his opponent average of .205 was third best among the crop of free agents.

Pitching is the A’s top priority, and they are also looking to move infielder Brett Lawrie. Teams that have some interest, according to industry sources, are Cleveland, Detroit and the White Sox. The Yankees are thought to be a potential fit if they decide to add a second baseman.

Susan Slusser is a San Francisco Chronicle staff writer. E-mail: sslusser@sfchronicle.com Twitter: @susanslusser