The return package for Houston was not initially reported.
According to McTaggart, McCullers Jr. had a no-trade clause, meaning he had to approve the deal to Milwaukee.
McCullers Jr., 32, has made only eight appearances this season after suffering a right rotator cuff impingement in mid-May. He is on the 15-day injured list and has made three rehab starts at triple-A.
Over his eight games with the Astros this year, the Tampa, Fla., native has a 6.86 ERA and 1.525 WHIP, with 43 strikeouts in 39.1 innings pitched.
Drafted by the Astros in the first round, 41st overall, in 2012, McCullers has spent his entire nine-year career in Houston. He won World Series titles with the team in 2017 and 2022, and was named an all-star in 2017.