For general manager Jon Horst, Budenholzer had always been his top target, and a finalized contract is expected to be completed soon, league sources said.
The person confirmed the agreement with the former Atlanta Hawks coach on Wednesday.
Toronto fired Dwane Casey, who may win Coach of the Year, after Cleveland swept the Raptors in the second round.
Putin opens controversial 19km bridge to Crimea
Ukraine, which along with most of the worldwide community has not recognised Russia's annexation of Crimea, condemned the project. The bridge capacity is 40,000 cars and 47 pairs of trains per day, 14 million passengers and 13 million tons of cargo per year.
With Budenholzer having joined the Bucks, the Raptors' search for a new coach continues onward. He pulled out of contention for the Suns job after several conversations with ownership and management. Budenholzer had been the Hawks' president of basketball operations - until surrendering front-office control with the arrival of general manager Travis Schlenk in the spring of 2017.
Budenholzer was named the NBA Coach of the Year in 2014-15, but agreed to part ways with the Hawks after the season.
Budenholzer went 213-197 in his five seasons in Atlanta, leading the team to four playoff appearances before this season's 24-58 record.
The Bucks lost to the Celtics in seven games in the first round of the playoffs.