According to Ken Rosenthal of the Athletic, the White Sox also sent $1 million to cover part of the $3.25 million owed to the reliever for the rest of this season.
The 34-year-old is in the midst of one of his best seasons in years, producing a 2.56 ERA in 40 appearances with 16 saves. He also has 49 strikeouts in 38.2 innings.
Soria is in the last season of a three-year deal initially signed with the Kansas City Royals in 2016, but he struggled with consistency over the past two years and was traded to Chicago in January as part of a three-team deal.
He has been much better in his new location, increasing his strikeout rate and lowering his walk rate while also working in more high-leverage situations.
While he isn't quite as effective as he was earlier in his career as a two-time All-Star who earned Cy Young and MVP votes in 2010, he should be another quality piece to a Milwaukee bullpen that was already one of the best in the league.
The Brewers entered Thursday with a 3.31 bullpen ERA, which ranks fifth best in the majors, per ESPN.
While the team is currently 2.5 games behind the Chicago Cubs for first place in the NL Central, this type of move could help secure the first playoff berth for the franchise since 2011. Soria also has a $10 million option for 2019 if he remains effective with the Brewers, but a $1 million buyout if they decline the option.
Meanwhile, Chicago adds to one of the top farm systems in baseball.
Kodi Medeiros was considered the Brewers' No. 13 prospect by MLB.com, and he's pitching well in Double-A with 3.14 ERA in 103.1 innings. The 22-year-old left-hander could be a quality starter in the majors within the next couple of years.
Wilber Perez is more of an unknown as he's still pitching in the Dominican Republic, but the 20-year-old righty possesses quality upside to help complete this deal.