CHICAGO -- The lone known certainty about the relationship between the White Sox and Paul Konerko is that the team did not offer the free agent a qualifying offer of $14.1 million by 4 p.m. CT on Monday, with the same decision given to fellow free agent Gavin Floyd.

Whether the team captain and important organization staple returns for a 16th and quite possibly final year on the South Side has yet to be determined or at least has yet to be made public. It's an extraordinarily tough quandary born out of the highest level of mutual respect.

As recently as May 27, 2012, Konerko was hitting a robust .399. It was made all the more impressive by the fact that the first baseman does not count speed among his skillset.

His average from that date moving forward checks in at .249 over 837 at-bats, with 27 homers and 96 RBIs. Konerko had just five homers and seven doubles over the final three months of the '13 campaign covering 218 at-bats. Of course, Konerko has been beset by injuries from a floating bone chip in his left wrist that was surgically removed at the end of last season to a balky back that he dealt with in '13.

Those injuries are more along the line of maladies coming from the intense work put in behind 17 years of Major League success. Then, there were the more freak injuries such as getting hit in the face by a Jeff Samardzija pitch or a concussion caused by taking an elbow to the head as the Royals' Jarrod Dyson tried to beat out a play at first.

Konerko would not return just to take a victory lap across the 2014 campaign. That does not fit the style of this understated leader. The man who batted .253 with three homers and 10 RBIs over his final 83 at-bats last season has to decide if he also can find a desired level of comfort at the plate while playing what almost certainly figures to be a part-time role.

Other teams can officially sign Konerko as of Monday night after 11:01 p.m. CT, most likely an American League squad with which the 37-year-old could split time between first base and designated hitter. With a young family of three children, Konerko would probably look for a team located somewhat close to his Arizona home and with a Spring Training locale in Arizona.

But judging Konerko's value to the White Sox goes well beyond on-base percentage, slugging percentage, FWar or BWar, especially in this one-year scenario. Konerko has been an unofficial coach within the clubhouse, a veteran whom young position players and pitchers alike will seek out for advice on matters related to or around the game.

There was a moment early on in Spring Training this year when veteran Alex Rios spent about five or six minutes discussing his swings from that particular Cactus League game with Konerko before the two departed for the day. For what looks to be a young team in '14, a group that the White Sox hope will contend but aren't painting as a frontrunner, Konerko could play a valuable role beyond his offense or defense.

Having a roster with Konerko, Adam Dunn and Jeff Keppinger becomes hard to envision with just four bench spots open and the White Sox looking to get more athletic and versatile, although Keppinger has the ability to play across the infield. That problem could be solved by going with one fewer pitcher at the outset, instead of the normal 12, or by using Leury Garcia as both a backup infielder and outfielder as the final reserve.

One of these three not being part of the upcoming group seems to make the most sense. With Keppinger owed $8.5 million over the next two years and Dunn having $15 million due in the finale of his four-year deal, either move seems unlikely without the White Sox picking up a sizable chunk of salary.

Maybe Konerko already has an idea in mind and is waiting for the White Sox to confirm his thoughts. Maybe the White Sox feel the same and hope Konerko's plans match their outline.

Other prominent players have left the White Sox, and more will do the same in the future. As Konerko told MLB.com back in April, he received an early indoctrination into the business side of the game by being traded twice at the age of 21.

"Right off the bat, I learned this is a business and this is, you know, it's run like a business," Konerko said.

This decision is different than other exits by key cogs. It won't be rushed by the White Sox, and Konerko's hand won't be forced.

"I'm committed to let Paulie go out. He's earned the right to go out however he wishes," said White Sox executive vice president Ken Williams. "So, any comments that I may have one way or the other I think are unfair and inconsistent with that desire."