Shaw had put together two impressive seasons for the Brewers after they acquired him three years ago from the Red Sox in a multiplayer deal that sent reliever Tyler Thornburg to Boston.
Shaw, 29, batted .273 with 31 home runs and 101 RBIs in 2017, then followed that with a .241-32-86 line in 2018.
But Shaw, who has started at both second and third base in Milwaukee, slumped to .157 with just 7 home runs and 16 RBIs last season.
Milwaukee also did not tender contracts to pitchers Junior Guerra, Jimmy Nelson and Alex Claudio, and infielder Tyler Saladino. Nelson struggled after missing nearly two years following surgery on his right shoulder. Milwaukee projects to save around $15 million in salary in 2020 by cutting the five players.
The Brewers and left-hander Ben Gamel agreed to a $1.4 million, one-year deal to avoid arbitration. The contract includes a team option for 2021 at $2.55 million with no buyout. Shortstop Orlando Arcia also agreed to a $2.2 million, one-year deal to avoid arbitration.
Milwaukee has parted with a number of key players this offseason. After catcher Yasmani Grandal left as a free agent for a $73 million, four-year contract with the Chicago White Sox last month, infielder Mike Moustakas agreed to a $64 million, four-year deal with Cincinnati on Monday, league sources tell ESPN's Jeff Passan.
Information from The Associated Press was used in this report.