General manager Danny Ferry called on his past experience with San Antonio to select Budenholzer to replace Larry Drew as coach. Drew's contract expires in June following three seasons as coach.
The Hawks have scheduled a news conference for Budenholzer on Wednesday in Atlanta.
The Hawks lost to Indiana in the first round of the playoffs. It was the team's sixth straight postseason appearance but Ferry is expected to continue his dramatic makeover of the team after trading Joe Johnson and Marvin Williams in his first year as general manager.
The Hawks say Budenholzer will continue serving as the Spurs' top assistant through the NBA finals. The Spurs completed a four-game sweep of Memphis in the Western Conference finals on Monday.
Ferry came to the Hawks last year after two years as vice president of basketball operations for the Spurs. With Budenholzer, Ferry now has his own coach after inheriting Drew.
"He has an incredible basketball acumen and has a keen awareness of the league and what it takes to be successful," Ferry said Tuesday in a statement released by the team. "His experience and four championships over the last 17 years provide a tremendous foundation for his leadership of our team."
Ferry, contacted by The Associated Press on Tuesday, declined to make additional comments until Wednesday's news conference.
Ferry calls Budenholzer "Bud" and "Coach Bud." Ferry was believed to have Budenholzer on his short list of candidates from the start of his search.
Budenholzer, 43, spent 19 years with the Spurs, including 17 seasons as an assistant. He was the top assistant for coach Gregg Popovich the last six years.
"I have been extremely fortunate to be a part of the San Antonio Spurs organization for the last 19 years," Budenholzer said in a statement released by the Hawks. "I knew it would have to be a tremendous situation for me to leave and clearly coming to Atlanta as the head coach of the Hawks is perfect for me.
"Ownership's commitment to taking this organization to the next level and creating a unique and special culture, partnering with a general manager like Danny Ferry who I have great respect for, and building a roster that has terrific potential because of the existing core and the ensuing flexibility presents a rare and uniquely positive opportunity."
The Hawks' new coach is looking for his fifth NBA championship ring this year. The Spurs won NBA titles in 1999, 2003, 2005 and 2007.
Budenholzer began his time with the Spurs as the team's video coordinator for two years before being named an assistant coach.
He will leave the Spurs with Popovich's blessing.
"I couldn't be happier for Mike for many reasons," Popovich said. "As anyone who's been part of this program knows, he has been more of a co-head coach than an assistant for a long time. His knowledge of the game as well as his ability to teach and develop relationships with players are all special. I will miss him a great deal both professionally and personally and am confident that he and Danny will make a great team as the future unfolds."
San Antonio guard Tony Parker said Budenholzer "has been extremely important to our success in San Antonio."
"He is a great coach," Parker said. "I think the Hawks made an excellent choice and I'm very happy for Coach Bud."
Led by their new management team of Budenholzer and Ferry, the Hawks could be factors in the free-agent market.
Center Al Horford is the lone starter with a guaranteed contract for next season. The only other players with guaranteed contracts are guard Lou Williams, who is recovering from a season-ending knee injury, and rookie guard John Jenkins. Point guard Jeff Teague can become a restricted free agent.
Longtime starting forward Josh Smith tops Atlanta's long list of unrestricted free agents.
If only the three players with guaranteed contracts return, the Hawks could have about $40 million under the cap to spend in free agency.
Drew could not be reached for comment. He endured what was essentially a lame-duck season after the team renewed the option on his contract last summer but didn't give him an extension.
Drew was 128-102 in three seasons.
"I thought we had a really good season given what our circumstances were. They weren't the best," Drew said after the season. "... The one thing I will say is I've had a great run here."
Drew began his time with the Hawks as an assistant under previous coach Mike Woodson. He moved up in 2010 after Woodson was hired. The Hawks advanced to the second round of the playoffs in Drew's first season but took first-round losses in 2012 and 2013.