In return, the Grizzlies will get guard Sam Merrill and two future second-round picks, sources said.
Allen, the 25-year-old former Duke standout, set career highs in points (10.6), rebounds (3.2) and assists (2.2) per game this season for a Memphis team that advanced from the play-in tournament to become the eighth seed in the Western Conference playoffs.
The 6-foot-4 guard also shot 39.1% from 3-point range while averaging 25.2 minutes per game. Allen's addition may help the NBA champion Bucks fill the long-distance void left by Bryn Forbes, who declined his player option and became an unrestricted free agent.
Merrill, a second-round pick in last year's draft, averaged 7.8 minutes per game while appearing in 30 games for Milwaukee, starting two.
Also Friday, the Bucks announced that guard George Hill is returning to the team after playing for it from 2018-20. The Bucks also officially announced the signing of forward Semi Ojeleye and the re-signing of forward Bobby Portis.