By Bruce Levine-

PHOENIX (CBS) — As these general managers meetings come to a close in Arizona, the White Sox appear to have done their due diligence, trying to find a closer and starting pitcher for next season.

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The White Sox met earlier this week with agent Mark Rodgers, who represents reliever Andrew Miller, sources confirmed to 670 The Score. It appears the parameters of a long-term contract for Miller will be four years and around $35 million to $40 million. A 29-yer-old left-handed setup man, Miller also has closer stuff. Splitting time between the Red Sox and Orioles this past season, he had one of the most dominant seasons in baseball last, striking out an incredible 103 batters in 63 2/3 innings. He had a 2.02 ERA and an 0.80 WHIP, the lowest of any setup man in the American league.

While Rodgers met with multiple teams here in Phoenix, the White Sox would be a perfect fit for the hard-throwing Miller.

“There is no direct science to the free agent market,” White Sox general manager Rick Hahn said, not directly addressing Miller. “When a player doesn’t sign with you, it sometimes knocks loose a player that was not available before for trade. It is certainly more art than science about when to strike. When it makes sense and lines up accordingly, we will go either route to improve our club.”

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Miller is exactly what the White Sox need to improve several areas in the bullpen. Chicago needs a great setup man, a possible closer alternative and a left-handed arm at the back end of games to match up in pressure situations. After being touted as a starter in the big leagues, Miller became a full-time reliever in 2012 with the Red Sox. Over the past two seasons, he has developed into a lockdown late-inning star.

Boston traded Miller to Baltimore to at the July 31 deadline with free agency looming for the pitcher.

“Back-end bullpen help is a high priority for us,” Hahn said. “We have made it no secret that the performance of the bullpen last year was unacceptable. This is certainly an important area of need for us. We are exploring all sorts of different options.”

Last season, Chicago lost potential closer Nate Jones to elbow surgery, and he’ll be out until at least mid-summer 2015. Jake Petricka was the closer at the end of 2014 for the Sox. He had 14 saves and a 2.96 ERA.

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Bruce Levine covers the Cubs and White Sox for 670 The Score and CBSChicago.com. Follow him on Twitter @MLBBruceLevine.