By Bruce Levine–

CHICAGO (CBS) — The Cubs added to their rotation with the signing of right-hander Tyler Chatwood to a three-year deal Thursday, but there are still more valuable innings that need to be filled with Jake Arrieta and John Lackey shopping their talents in free agency.

With that in mind, the Cubs are continuing their strong push to sign free agent right-hander Alex Cobb, sources said. The rumors on a Cobb-Cubs marriage have been out there for several months, and with the Winter Meetings starting Monday, Chicago would like land Cobb as quickly as possible to avoid any traps and new suitors that may await in the next week. Agreeing to a deal with Cobb before Monday would also give the Cubs more leverage during the Winter Meetings because they wouldn’t be as desperate to fill a rotation spot.

In the Cubs can add the 30-year-old Cobb on a multi-year deal in the coming days, they would have five starting pitchers under team control for three more seasons. Jon Lester, Jose Quintana, Kyle Hendricks and Chatwood all have contracts that run through 2020, and any deal for Cobb would be at least that long.

The Yankees are believed to be the main competition to the Cubs for Cobb, who has spent his entire big league career with the Rays and who went 12-10 with a 3.66 ERA and 1.22 WHIP. A four-year deal is assumed to be the end game for Cobb’s services.

Adding Cobb would give the Cubs some breathing room as they look to develop young pitchers at the lower levels of the minor leagues. Some of those top prospects are right-hander Jose Albertos, right-hander Adbert Alzolay, right-hander Oscar De La Cruz, right-hander Alex Lange and left-hander Brendon Little.

Cobb has a great relationship with new Cubs pitching coach Jim Hickey, who mentored Cobb for six season in Tampa Bay. Joe Maddon was also Cobb’s first manager in the big leagues.

The Cubs are determined to add more pitching to their rotation and bullpen.

“We have plenty more work to do,” Cubs president of baseball operations Theo Epstein said after signing Chatwood. “We are certainly not done. The focus of the winter has been pitching. We need to continue to add to the rotation and retool the bullpen to a certain extent.”

The Cubs need to fill a void at closer as Wade Davis has hit free agency. A return to Chicago for Davis is still possible, and veterans Addison Reed and Brandon Morrow are high on the Cubs’ wish list.

And of course, the Cubs will continue to talk with Davis and Arrieta.

“Jake is still a free agent and has done great things here,” Epstein said. “Until it reaches a point he signs somewhere else, I am sure we will be in dialogue with (agent Scott Boras). We always talk with him at the Winter Meetings. I am sure we will catch up. Wade, the same thing.”

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