The Wizards will absorb Dudley's contract into an existing trade exception, and the Bucks will get back a future second-round pick. The deal is on course to be completed July 9.
Dudley shared the news on Twitter:
He also said goodbye to Milwaukee:
Milwaukee!!! I know it's was only there for 1 yr but what a yr!!! From 17 wins to 41 and the playoffs!! We started the foundation #goodluck— Jared Dudley (@JaredDudley619) July 3, 2015
On the last day before free agency began, Dudley opted into the final year of his contract at $4.25 million for the 2015-16 season.
The 37-year-old Pierce was particularly terrific in the playoffs for the Wizards, giving the young backcourt of John Wall and Bradley Beal a veteran who was comfortable taking big shots late in big games. Pierce helped the Wizards sweep Toronto in the first round then delivered a series of clutch shots during their six-game elimination against Atlanta in the Eastern Conference semifinals.
Washington can take on Dudley's salary without sending any players back in the deal thanks to a trade exception it picked up as part of a swap that sent Andre Miller to the Sacramento Kings for Ramon Sessions at the trade deadline in February.
Dudley rebounded following a disappointing stint with the Clippers, who dealt the swingman to the Bucks in a cost-cutting move after the 2013-14 season. After shooting a career-low 43.8 percent with the Clippers, he connected at a 46.8 percent rate in his first season with the Bucks, though he averaged just 7.2 points per game.
Dudley's real value comes from his ability to defend multiple positions. When Milwaukee elected to go small, the 6-foot-7 small forward would rotate to play power forward.
Earlier Thursday, the Bucks agreed with free-agent center Greg Monroe on a $50 million, three-year contract. Dudley became expendable for the Bucks when they got Greivis Vasquez from the Toronto Raptors to give them a veteran shooter off the bench.
The Associated Press contributed to this report.