ESPN.com's Adrian Wojnarowski first reported the news Wednesday night.
Wojnarowski added Budenholzer agreed to a four-year contract.
Gery Woelfel of the Racine Journal Times then reported the coach was one of the favorites to fill the Bucks opening, while Marc Stein of the New York Times noted the Raptors' "strong interest" in him.
Wojnarowski noted Wednesday that Toronto met with Budenholzer but never extended an offer.
The 48-year-old reportedly has now been hired by Milwaukee, where he will try to replicate the early success he found in Atlanta.
Despite the struggles of the past year during the start of a rebuild, Budenholzer finished his time with the Hawks with a 213-197 record in the regular season with four playoff appearances in five years. The 2014-15 campaign featured a 60-22 record where the team earned the No. 1 seed in the Eastern Conference along with a trip to the conference finals.
He was named the NBA Coach of the Year that season.
Prior to that, Budenholzer spent 17 seasons as an assistant with the San Antonio Spurs, helping the team win four titles in that span.
His experience on NBA sidelines could help him once again get the most out of his players in the new location.
He will have plenty to work with in Milwaukee, a team led by one of the best young players in the NBA in Giannis Antetokounmpo, with whom Budenholzer reportedly had breakfast Wednesday morning, per ESPN's Zach Lowe. Khris Middleton, Eric Bledsoe, Malcolm Brogdon and more will also return for at least another year, hoping to find more success than either Jason Kidd or Joe Prunty did last season.
With a franchise that hasn't won a playoff series since 2001, Budenholzer has a low bar but should have a lot of expectations for next season and beyond.