Rios has not been acquired which means he's not officially a member of the Rangers. The Rangers must work out a deal with the White Sox in order for Rios to head to Texas. Steve Adams of MLB Trade Rumors explains more into how the waivers work.
Because [Rios] is on revocable waivers, the White Sox have 48 hours to work out a trade with the claiming team or pull him back off waivers. Chicago could also let the claiming team have Rios for nothing and be content to dump his salary.
The waiver priority is also determined by the league and record. Teams in the American League have the first priority.
There was much speculation on Twitter to whether or not Rios would land with the Rangers or Pirates, both teams looking for an extra bat to make a deep playoff run. The Rangers, who lost Nelson Cruz to a 50-game suspension, were looking at acquiring Rios before the 2013 MLB Trade Deadline.
Rios is a corner outfielder who currently is hitting for a .277 average with 12 home runs and 55 RBIs. For the White Sox, Rios played right field and batted third. Joining the Rangers would mean a reunion with former teammate A.J. Pierzynski.
Back before the trade deadline, Pierzynski contacted Rios about the Rangers looking into trading for the outfielder.
The Rangers are now tied for first place in the AL West. They've won nine of their last 10 games and travel to Houston Friday to play four games with the Astros. If the Rangers can strike a deal, Alex Rios could make his debut this weekend in Houston.