Nash took charge of the Nets at the start of the 20-21 season, leading the Nets to the Eastern Conference semi-finals and a 48-24 regular season record.
Brooklyn were beaten by the eventual champions Milwaukee Bucks in overtime of Game 7 in the east semis.
In his second season in charge, Nash led the Nets to a 44-38 record as his side were swept by the Boston Celtics in the first round of the playoffs.
The off-season was filled with debate over Nash's future, with star player Kevin Durant asking for a trade.
Durant's exit did not come to fruition after it was reported that he wanted out or for Nash and general manager Sean Marks to be fired.
It has been a difficult first coaching job for Nash, who has had to deal with lots of issues surrounding Kyrie Irving, who was not available for home games last season until March due to his vaccination status.
The Nets also acquired Ben Simmons from the Philadelphia 76ers in February as part of the James Harden trade.
Simmons did not play at all last season due to a back injury but has played all six games to start the new campaign, with Brooklyn sat 12th in the Eastern Conference with two wins.
The Nets face the Chicago Bulls on Tuesday evening, with Jacque Vaughn serving as acting head coach.
News of Nash's exit was initially broken by ESPN's Adrian Wojnarowski before The Athletic correspondent Shams Charania also confirmed the news.
After the 'mutual' decision was made, Wojnarowski confirmed that the Nets are looking at former Celtics boss Ime Udoka, who was suspended for the entire 22-23 season for violations of team policies.
The Cs will apparently not stand in Udoka's way if he wishes to join a new franchise.
Udoka led Boston to the NBA Finals last season, where they were beaten by the Golden State Warriors in six games.
Brooklyn are also said to be looking at former Utah Jazz head coach Quin Snyder, among others.
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