The White Sox will receive two prospects -- left-hander Kodi Medeiros and right-hander Wilber Perez -- from the Brewers, who will also get $1,064,516 to cover part of the $4,145,161 remaining in the guaranteed portion of Soria's $25 million, three-year contract. Soria has a $9 million salary this year, and his deal includes a $10 million mutual option for 2019 with a $1 million buyout.
"What we're doing with adding Soria is just adding another guy to get outs down in the bullpen,'' Brewers manager Craig Counsell said. "We'll just kind of continue to add to a group that's been really, really good this year. He's having a very nice season. He's performed at the back end of a bullpen. But he'll get outs for us, like we consider all of those guys down there as just out-getters.''
Counsell added that Soria won't be the team's closer.
"This is the price of poker," Brewers general manager David Stearns told MLB.com. "We view Soria as one of the better available arms on the market this year, and he fits our team well."
Soria, 34, had emerged as Chicago's closer, compiling 16 saves this season. He had a 0.74 ERA with 32 strikeouts over his last 23 games dating to May 21. The White Sox had acquired the veteran reliever from the Kansas City Royals as part of a three-team trade in January.
"I'm just going there to try to help them in whatever situation they feel I'm fit for," Soria told MLB.com. "And I'm going to be ready for them for every type of scenario they can put me."
The two-time All-Star was 4-3 with one save and a 3.70 ERA in 59 games for the Royals last season. He has 220 career saves and 662 strikeouts in 613 games since entering the majors in 2007.
"This is a guy who has effectively handled some very close games and high-leverage situations this year," Stearns said. "We feel great bringing him into the fold."
Milwaukee trailed the Chicago Cubs in the NL Central by 2.5 games entering play Thursday.
Medeiros, 22, is 7-5 with a 3.14 ERA in 20 appearances (15 starts) for Double-A Biloxi this season. He was the Brewers' first-round pick (12th overall) in the 2014 draft.
Perez, 20, is 5-1 with a 2.01 ERA in eight starts with Dominican Summer League Brewers. The native of the Dominican Republic was signed by the Brewers as an international free agent in 2017.
The Associated Press contributed to this report.