The White Sox latest loss in Minnesota was the perfect backdrop for the team unofficially raising the white flag.
For players to be eligible for the postseason with a new team, transactions had to be completed before midnight on Monday.
Thome is batting .252 with 23 homers and 74 RBIs in his fourth season in Chicago. The five-time All-Star is 12th on the all-time home run list with 564.
ESPN The Magazine's Buster Olney reported earlier on Monday that the White Sox had sent a memo to teams around the league saying veteran players were available for trade. Thome, Jermaine Dye and Scott Linebrink were said to be on that list.
It's not immediately clear what role Thome will fill for the Dodgers. He has been exclusively a designated hitter the past two seasons and hasn't played first base -- the position he played earlier in his career -- since appearing in one game there in 2007. He hasn't played first base full time since 2004.
"I'm not saying you won't see Jimmy at first, but I want to talk to him," Torre said.
James Loney is the Dodgers regular first baseman, but he has only nine homers.
Dodgers general manager Ned Colleti envisions Thome playing a role similar to the one Matt Stairs filled for the World Series champion Phillies last year -- he hit a key home run against the Dodgers in the NLCS.
"We're not brining him over here to play first base," Colleti said. "We're bring him here to come off the bench and be a great influence in the clubhouse. He's one of the true great guys in the game.
"In fact, the night before the deadline he called me. ... He just said: 'I just want to be honest with you. I'd love to come. I want to help you guys any way I can. But playing first base is not something I'm going to be able to do -- maybe in an emergency situation, perhaps.'"
Contreras leads the American League with 13 losses against only five wins in 21 starts. He has a 5.42 ERA this season.
The Dodgers will receive Thome and cash for minor league infielder Justin Fuller. The Rockies receive Contreras and cash for minor league pitcher Brandon Hynick.
Hynick is 10-9 with a 3.83 ERA and two complete games in 26 starts this season with Triple-A Colorado Springs.
After another loss to the Twins in Minnesota, the White Sox are third in the AL Central, six games behind the Tigers and 2 1/2 behind the Twins. They have lost eight of nine overall and 15 of their last 17 to the rival Twins in the Metrodome.
Information from The Associated Press was used in this report.