The White Sox will send $1 million to Texas toward Rios' salary, sources told ESPN, while receiving either a player to be named later or cash considerations.
"Well, I believe it's going to be refreshing for myself," Rios said. "I'm going to go to a team that is fighting for a playoff spot. I believe it's going to be refreshing. I believe they are hoping to win their division and go to the playoffs so let's see what happens."
The player to be named later is expected to be Leury Garcia, sources said.
The Rangers put in a waiver claim for Rios on Thursday. Rios won't play Friday, according to Rangers manager Ron Washington, but when he is inserted into the lineup, he'll bat sixth and play right field.
"He's a good fit for us," Rangers general manager Jon Daniels said. "We had interest in him prior to learning of Nelson's suspension and continue to have interest him and we were able to work out a deal."
Rios' contract allowed him to block trades to six teams, but Texas was not on the list. He is owed $13 million next season and the remaining portion of his $12 million salary this season. He has a $2 million buyout for 2015. A trade clause in his contract boosted his 2014 and 2015 salary, as well as the 2015 buyout.
"This is for us another step in the process of retooling this club," said White Sox GM Rick Hahn. "It's difficult to let a player like Alex Rios go. He, in my opinion, has been outstanding for us on and off the field during his stay in Chicago. A pure class act."
The two teams had many discussions before the non-waiver trade deadline without coming to an agreement.
"It was a great ride," Rios said of his time in Chicago. "I enjoyed my whole time here. I had many great moments here. So, I'm going to miss this place."
Rios is sixth in the American League with 26 steals and he's tied for fifth in the league with eight outfield assists.
"He's a complete player," Washington said. "He's solid outfielder. He's a tremendous baserunner. He just gives us completeness. That's what we needed."
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 could work a deal only with the Rangers, significantly reducing a potential return. Rios wasn't surprised by the trade.
"With the situation that we were in, it wasn't too much of a surprise," he said. "They were trying to get rid of salary or whatever they wanted to do. I think many people expected this to happen. You know what, it's all good."
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.
With Rios on the move, it could open the door to get newly acquired prospect Avisail Garcia some playing time at the big league level. Garcia, who was acquired from the Detroit Tigers in a three-team trade July 30 that sent Jake Peavy to the Boston Red Sox, was recalled Friday.
"This opens an opportunity for playing time for Avisail Garcia who will be traveling to Chicago right now and hopefully will be here for tonight's game and that's a big consideration for us as we start to retool this club and bring in that next group of guys and get them going on their major league career and development," Hahn said.
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.
Information from ESPN Insider Jim Bowden, ESPNDallas.com's Todd Wills and ESPNChicago.com's Bruce Levine and Doug Padilla was used in this report.