ESPN reported that the Mavericks will also get guard Max Christie and a 2029 first-round draft pick, while the Lakers will also receive Maxi Kleber and Markieff Morris.
The Utah Jazz, the third team in the deal, will get Lakers guard Jalen Hood-Schifino.
Reports of the trade surfaced shortly after the Lakers - with Davis sidelined by an abdominal muscle strain - beat the Knicks 128-112 in New York.
“According to people inside the Lakers’ locker room not authorised to speak publicly, the trade was a complete shock,” the Los Angeles Times reported.
Doncic has been sidelined since Christmas with a calf injury.
He is averaging 28.1 points, 8.3 rebounds and 7.8 assists per game for the Mavericks, where he has played since he was drafted third overall by the Atlanta Hawks in 2018 and immediately traded.
Davis, 31, has spent six seasons with the Lakers. A nine-time All-Star, he won a title with the Lakers in 2020.
He is averaging 25.7 points, 11.9 rebounds and 3.4 assists per game this season.