Sam Amick of The Athletic reported the news Thursday and noted Tyronn Lue is also expected to receive a "serious look" from the Rockets if D'Antoni leaves the organization.
Last May, Houston owner Tilman Fertitta said he made a "great extension offer" that included a $5 million base salary on a one-year deal with a $1 million bonus for each postseason series won.
"It was one of the better base salaries in basketball but also had the highest incentives in basketball. Mike could make $4 million in incentives. Business is business," Fertitta told reporters. "I'm a very incentive-based guy. If you perform, I want you to do unbelievable."
D'Antoni rejected the proposal and decided to coach out the final season of his previous contract.
"I think we're going to have a great year," he said. "And we'll go from there."
Amick suggested D'Antoni could be interested in the New Orleans Pelicans' head coaching job if the team moves on from Alvin Gentry.
D'Antoni, who'd inherit budding superstar Zion Williamson under that scenario, previously worked with Pelicans executive vice president of basketball operations David Griffin while with the Phoenix Suns.
The 69-year-old West Virginia native owns a 672-526 regular-season coaching record across 16 years with the Rockets, Suns, New York Knicks, Los Angeles Lakers and Denver Nuggets. His teams have posted a more modest 49-49 postseason record and haven't made an NBA Finals appearance.
Van Gundy previously coached the Rockets from 2003 through 2007. The team made the playoffs three times in four years but failed to advance beyond the first round. He'd previously led the New York Knicks from 1995 until early in the 2001-02 season, highlighted by a trip to the 1999 NBA Finals.
More recently, he's worked for Team USA to help the national team qualify for the 2019 FIBA World Cup and 2021 Summer Olympics. His efforts drew praise from San Antonio Spurs head coach Gregg Popovich, who's set to lead the U.S. team in the Olympics.
"He's better than most of us as a coach," Popovich said, per Newsday's Steve Popper. "Most people know that. That's not always what gets you hired. It's circumstances and those sorts of things; personal situations and timing has to be right and all that. But he's certainly somebody who would make a franchise better, without any doubt."
Meanwhile, Lue is generating a lot of buzz about being a coveted target during the offseason after spending the 2019-20 campaign as an assistant with the Los Angeles Clippers.
Chris Mannix of Sports Illustrated reported Wednesday the former NBA point guard is set to become a "leading candidate" for "several" jobs as they open in the coming weeks.
Lue spent parts of four seasons leading the Cleveland Cavaliers, beginning in the middle of the 2015-16 campaign, and helped lead the franchise to its first championship in 2016.