"I watched all the reports online and television " Rios told ESPNChicago.com. "My agent said he heard nothing was close as of last night."
"We love it here in Chicago," Rios said. "My wife doesn't want to go anywhere, but some of that is out of our control. We have young kids and we have enjoyed our summers here. I have great teammates and the White Sox have treated us very well."
Rios' contract allows him to block trades to six teams, but Texas is not on the list. Rios is owed $12.5 million next season and the remaining portion of his $12 million salary this season. He has a $1 million buyout for 2015. If he is traded the buyout goes up to $2 million and his contract goes up $500,000 for 2014 and '15.
The two teams had many discussions before the non-waiver trade deadline without coming to an agreement.
Teams typically put players through waivers at some point in August but can pull them back if claimed. Each player who makes it through revocable waivers unclaimed is eligible to be traded to any team just like he was before the July 31 non-waiver trade deadline.
But with Texas claiming Rios, the White Sox can work a deal only with the Rangers, significantly reducing a potential return.
The final option for the White Sox is to pull Rios off waivers and keep him. If a player is claimed and a deal is not made within 48 hours, he can't be put on waivers again for 30 days. That would render him useless for playoff rosters, which must be submitted by Sept. 1. The White Sox could deal Rios in the offseason or move him before next season's July 31 non-waiver trade deadline.
The White Sox were fielding trade offers for Rios before the July 31 deadline but never found a match. The Rangers and Pittsburgh Pirates were reportedly in the mix. A league source said negotiations with the Rangers ended when the White Sox wanted 20-year-old second base prospect Jurickson Profar.
The Rangers need offensive help after losing Nelson Cruz to suspension in the wake of the Biogenesis investigation. Rios entered Friday hitting .277 with 12 home runs and 55 RBIs.
If Rios is moved, it would open the door to get newly acquired prospect Avisail Garcia some playing time at the big league level. Garcia was acquired from the Detroit Tigers in a three-team trade July 30 that sent Jake Peavy to the Boston Red Sox.
White Sox general manager Rick Hahn said he would want Garcia on the major league roster only if there was an opportunity for him to play every day. The corner outfielder has been on a tear since his arrival at Triple-A Charlotte with 10 hits in 24 at-bats (.417), including a home run and nine RBIs, entering Friday.
ESPNChicago.com's Doug Padilla contributed to this report.