Budenholzer met with Suns general manager Ryan McDonough and owner Robert Sarver early this week, but there was never traction on reaching a contract agreement as the week wore on, league sources said.
As the Suns kept interviewing candidates -- including David Fizdale and interim coach Jay Triano -- Budenholzer informed the Suns on Thursday that he would no longer be a candidate for the job, sources said.
The Hawks gave permission to Budenholzer to meet with the Suns, and would be open to allowing him to become a candidate for other head coaching openings, league sources said. Budenholzer has two years and $13 million plus left on his contract, with the Hawks headed into a full rebuilding process.
Phoenix met with Fizdale on Wednesday and Triano on Thursday and have had conversations with Steve Clifford, Kevin McHale, Frank Vogel and David Blatt, sources said. The Suns have also received permission to speak with San Antonio Spurs assistant James Borrego, league sources said.
Considered one of the NBA's best tacticians, Budenholzer had been the Hawks' president of basketball operations -- until surrendering front-office control with the arrival of new GM Travis Schlenk in the spring of 2017.
Budenholzer was the NBA's 2015-16 Coach of the Year, after a 60-victory regular season and a trip to the Eastern Conference finals. The Hawks reached the playoffs in four of his first five seasons. Atlanta dropped to 24-58 this season, with the organization's mandate to shed salary, gather draft picks and future assets.
Budenholzer spent 17 years as an assistant under Spurs coach Gregg Popovich before his hiring in Atlanta in 2013. He won four NBA titles as a member of Spurs' staff.