This Tuesday, July 7, 2015, file photo, shows Boston Celtics...

This Tuesday, July 7, 2015, file photo, shows Boston Celtics summer league head coach Jay Larranaga directing his team during the first half of an NBA summer league basketball game against the Philadelphia 76ers, in Salt Lake City. Credit: AP / Rick Bowmer

The Knicks’ list of candidates for their coaching vacancy continues to grow. They’ve interviewed Celtics’ lead assistant coach Jay Larranaga, according to a league source.

They received permission to speak to Larranaga and met with Brad Stevens’ top aide last week, the source said. Larranaga is the ninth known person the Knicks have interviewed or will interview.

But an even more intriguing development occurred with someone the Knicks talked to over the weekend.

The Hawks and head coach Mike Budenholzer mutually parted ways, the team announced Wednesday night. Budenholzer had two years and roughly $14 million remaining on his contract. He interviewed for the Knicks’ job Sunday.

A source labeled Budenholzer “a strong candidate.” Jerry Stackhouse, David Fizdale and Mark Jackson — three of the first people Knicks’ officials interviewed — are considered serious candidates as well.

But Budenholzer is the most accomplished one on the Knicks’ list. He might be even more attractive now that the Knicks wouldn’t have to give up any compensation — draft picks — if they hired Budenholzer.

He worked under Gregg Popovich for 17 years and was a part of four NBA-championship teams with the Spurs before taking over the Hawks in 2013. In Budenholzer’s second season, the Hawks won 60 games and he was named NBA Coach of the Year. Budenholzer also demanded Tim Hardaway Jr. play defense or he didn’t play in his two years with Atlanta.

The Hawks granted Budenholzer permission to interview for jobs so a split seemed to be a likely result. Budenholzer met with the Suns first, but he reportedly withdrew his name from consideration.

With two years left on his contract, it’s hard to imagine that Budenholzer would leave money on the table if he didn’t have something lined up. The Hornets and Magic are also looking for coaches, and the Bucks soon could be too.

Knicks president Steve Mills and GM Scott Perry, who are overseas this week to interview David Blatt and scout projected high-lottery pick Luka Doncic, are involved in an extensive coaching search to find Jeff Hornacek’s replacement.

They’re scheduled to sit down with Spurs assistant James Borrego after they return from Europe. The Spurs were eliminated from the playoffs Tuesday night. That interview could happen this weekend. The Knicks also have interviewed Mike Woodson and Kenny Smith.

After the Knicks fired Hornacek following a 29-53 season, Perry and Mills talked about wanting to find a strong defensive coach, who can relate well with the players and will hold them accountable.

They’re doing their due diligence as they try to find who they believe is the right person to lead the franchise through this rebuild. The fact that Mills and Perry are speaking to people who come from successful programs and don’t necessarily have ties to them — such as Larranaga and Budenholzer — would seem to be a good sign for a franchise that has won just one playoff series in 17 years.

Larranaga, 43, is a rising young coach, who has been with the Celtics for six seasons, the last four as Stevens’ top assistant.

Larranaga was considered for the Celtics’ job before they hired Stevens, one of the best coaches in the league. He has interviewed for head coaching jobs in Philadelphia and Memphis previously. The Celtics have reached the postseason for four straight seasons, and had the NBA’s top-rated defense

One of Larranaga’s strengths is a Knicks’ focus now: player development. He spent two seasons coaching the Erie BayHawks, which was the Knicks’ Developmental League affiliate before they purchased their own team in 2014. Jay’s father, Jim, is the head coach at the University of Miami.

If the Knicks decide on a younger, less-experienced head coach such as Stackhouse, Fizdale or Larranaga, they could bring in Woodson or Blatt as a lead assistant or associate head coach.

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