Having the third-highest active total of games played without reaching the playoffs at 1,395 would make moving to a contender an enticing proposition for Rios. But he enjoys Chicago, is under contract for 2014 and is not looking to leave.
Regardless of the outcome, Rios has put the trade thoughts out of his mind.
"Yeah. It gets old," Rios said. "When that date is arriving, you get a little anxious to see what your fate is going to be. Hopefully, after July 31, it is going to be normal.
"It's a good thing when people want you to be part of a good thing. But at the same time you have mixed feelings. You have spent so much time in an organization, you know pretty much everybody and you have to start over in another place. It's tough. But it is what it is."
Much like Jake Peavy, another trade candidate, Rios is surprised the White Sox are in this sell situation.
"Of course. I believed we had a good team to compete in this division or any division," Rios said. "We really believed it as a team. It's unfortunate what's going on and what has happened. Things happen in baseball."
Scott Merkin is a reporter for MLB.com. Read his blog, Merk's Works, and follow him on Twitter @scottmerkin. This story was not subject to the approval of Major League Baseball or its clubs.