The report said Irving has told the team he would like to move before Thursday's trade deadline or he will leave as a free agent in July.
Irving was selected as an NBA All-Star for the eighth time this season after averaging 27.1 points, 5.1 rebounds and 5.2 assists during the regular season so far.
The 30-year-old, who made his name with the Cleveland Cavaliers, is in his fourth year with the Nets after joining them from the Boston Celtics.
In November, Irving served an eight-game suspension after a social media post linking to a movie condemned as anti-semitic sparked a furore.
The report said Irving had recently turned down a contract extension offer from the Nets and that the Los Angeles Lakers, Phoenix Suns and Dallas Mavericks could be possible destinations for him in any move.
The Nets are fourth in the Eastern Conference with a 31-20 record.
