By Bruce Levine–

CHICAGO (CBS) — While baseball media types are anticipating that the Chicago White Sox will add another bat to their lineup, they could be pushing just as hard for another big arm in the rotation.

A strong top of the rotation headed by left-handers Chris Sale, Jose Quintana and Carlos Rodon is sill in need of right-handed starting fortification. The fact that Chicago needs another run-producing bat is just part of the offseason quest and puzzle for general manager Rick Hahn.

“We are going to continue to work,” Hahn said last weekend. “The marketplace still seems like it will be rather fluid for players right up until Opening Day.”

The fourth and fifth roles in the rotation — which were problematic in a disappointing 2015 season — will come down to a battle between Erik Johnson, John Danks and Jacob Turner. The White Sox’s 2015 first-round pick, Carson Fulmer is scheduled to pitch in the minor leagues and build innings viability during 2016.

The Tampa Bay Rays seem to be the target for many conversations between numerous clubs, including the White Sox and the Cubs. More than 10 clubs have asked Tampa about the availability of its young pitching.

A southern Illinois native, 25-year-old right-hander Jake Odorizzi appears to be the subject of many offseason talks. He has been traded twice since signing with the Brewers as the 32nd player chosen in the 2008 amateur draft. He was dealt to the Royals in a multi-player swap for Zack Greinke and traded to the Rays in a big deal for James Shields before the 2013 season.

Earlier this offseason, he was attached to rumors about the Cubs’ and Cardinals’ interest.

Both the Chicago teams have inquired about the Rays’ outfield and pitching depth. The Rays would seem to be in a mode of dealing young veterans like Odorizzi and outfielder Desmond Jennings for the right cache of good young prospects. This model for Tampa has worked well for years. The Rays appear ready to build for their next wave of strong contention in the big-market American League East, beyond the 2016 season.

This method isn’t tanking. It’s a form of surviving in a low-revenue market where the ticket base is always near the smallest in baseball. Volume additions in deals is what Tampa does best, such as in trades of players like the Shields and Matt Garza.

Odorizzi is under contract control for four more seasons. He will be arbitration-eligible in 2017 and a free agent after 2019. He has a 20-24 record over the past two seasons in 59 starts. He has a career 3.77 ERA and a 3-to-1 strikeout-to-walk ratio that makes him an attractive target for clubs with thin rotation depth.

Both Chicago teams remain active in player-procurement conversations.

“The season has not started yet,” Hahn reminded. “We are still a work in progress.”

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