According to ESPN's Adrian Wojnarowski, Boston has long had interest in the five-time All-Star (around the 2:23 mark):
"Boston has been hawking Anthony Davis for years. They always hoped that it would be—whether it's the end of this season or the beginning of next before the trade deadline—that they would gather up all those assets, all those picks Danny Ainge has, young players, and they'd be the team to be able to get Anthony Davis.
"But now you have L.A., and if they get shut out in free agency, they're going to have to take all their young players to try to use them to get Anthony Davis."
The Celtics were routinely linked to the 6'10" forward prior to last season's deadline.
The Athletic's Jay King also reported this December that Davis and Celtics guard Kyrie Irving—who is only signed through this season—have talked about the possibility of teaming up in Boston.
The 25-year-old Davis, signed through the 2019-20 season, is averaging 28 points, 12.4 rebounds, 4.7 assists, 2.8 blocks and 1.7 steals per game.
He will be eligible to sign a five-year, $235 million supermax next summer only with New Orleans, though the 2012 No. 1 pick switched his representation to Klutch Sports in September. (Klutch Sports also represents LeBron James, adding fuel to the AD-to-L.A. rumors.)
Boston came within one game of the NBA Finals last season despite injuries to both Irving and Gordon Hayward. Along with players on the current roster who could be attractive to the Pelicans—such as Jayson Tatum and Jaylen Brown—the Celtics have future assets they can include as well.