Wojnarowski added that Bickerstaff has "no assurances" long term but has the "opportunity to earn the job on a full-time basis." He also listed Jeff Van Gundy, Tom Thibodeau and Charlotte Hornets head coach Steve Clifford as potential options to land the full-time position.
Ken Berger of CBS Sports confirmed McHale's termination.
"The team was not responding to Kevin McHale," Rockets general manager Daryl Morey told reporters afterward.
"We believe in this team," Morey continued. "We're not in a good place. We're losing by double digits to teams we need to beat."
McHale spoke about his exit with Jonathan Feigen of the Houston Chronicle, saying he "didn't blame" the Rockets for the move but felt he would "have gotten it turned around." He hasn't decided what's next after leaving Houston.
The 57-year-old former Boston Celtics star signed a three-year contract extension in December 2014, and he guided the Rockets to back-to-back 54-plus win seasons. Still, he was unable to overcome a slow start to the current campaign.
Per Wojnarowski, the move wasn't easy for Houston:
Following Houston's loss to the Boston Celtics on Monday, McHale told ESPN.com's Calvin Watkins his team wasn't playing a strong brand of basketball: "I wouldn't say full effort all the time, no. We're hanging our head. Things aren't going our way and we hang our head. Bottom line, we haven't put together really good basketball, all year."
That coincided with point guard Ty Lawson suggesting there was a disconnect of sorts between the coach and his players:
Feigen backed that up following McHale's firing:
Rockets owner Leslie Alexander also spoke about the start to the season, telling reporters: "I've never seen my team...team play as poorly and [with] less effort. I'm not attributing it to the coach. I had to do something."
After head coaching stints with the Minnesota Timberwolves in 2004-05 and 2008-09, McHale became Houston's bench boss in 2011-12.
He led the Rockets to a winning record in each of his four full seasons at the helm, and he made the playoffs the past three years.
McHale took Houston to the Western Conference Finals last season, where the Rockets lost to the Golden State Warriors. That playoff run included overcoming a 3-1 series deficit against the Los Angeles Clippers in the Western Conference Semifinals.
This season, the Rockets have dealt with injuries to Patrick Beverley, Sam Dekker, Montrezl Harrell, Dwight Howard, Terrence Jones and Donatas Motiejunas, and they find themselves 11th in the hyper-competitive Western Conference.
But with James Harden and better luck on the injury front, this is still a team with loads of potential.
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