That is what ESPN insider Adrian Wojnarowski has said.
Davis will be eligible for a five-year super-max contract with the New Orleans Pelicans at the end of this season.
And though he does not have the player option to become an unrestricted free agent until July 2020, rejecting the deal – worth an incredible $235million – would put the Pelicans under pressure to trade away their talisman to avoid losing him for nothing a year later.
The 25-year-old is already a five-time All-Star and three-time All-NBA First Team player and is arguably the most all-rounded talent in the league.
And the Lakers have been linked with a move for Davis as they seek to surround LeBron James with another MVP-calibre player to boost their chances of winning a championship with him.
But they are expected to face fierce competition for the Celtics – who also have the pieces to tempt New Orleans into a trade.
And Wojnarowski warns that Boston have long had their eye on recruiting the towering power forward – who is averaging 28.1 points, 12.4 rebounds and 4.7 assists per game this season.
He said on the Woj and Lowe Podcast: “Boston has been hawking Anthony Davis for years.
“They always hoped that it would be, at the end of this season or beginning of next before the trading deadline, that they would gather up all those assets, the picks Danny Ainge has and the young players and they’d be the team to get him.
“But now you have LA, if they get shut out in free agency, they’re going to have to take all their young players to try to get Davis.
“There’s no deal that’s coming for New Orleans where Davis looks and says, ‘I think I can win a championship here.’
“If he stays with the Pelicans, it’s because of the $235m. They can make the team better but to honestly say you can see a championship contender, it reminds me a lot of KG [Kevin Garnett] in Minnesota at the end.
“Players keep passing on that supermax. Kawhi Leonard could have taken it in San Antonio and walked away from it.”
And Wojnarowski revealed the Pelicans are reluctant to make any trade deals unless they can bring in a player capable of becoming a franchise leader.
He added: “They thought they had a better team [this year] than the team that rolled Portland and got into the second round of the playoffs [last year] but they’ve had a lot of injuries.
“They haven’t been able to evaluate this group but they don’t have time to evaluate.
“They’ve got to see, ‘Can we piece together a deal out there?’ No-one has been more aggressive.
“They tried to get Jimmy Butler, offered their first-round pick and Mirotic to Minnesota, didn’t get him.
“I think their feeling there is, ‘We’re not going to bully-mortgage our future, give our pick away and have Anthony walk out the door unless we can get a difference-maker’.
“That’s what they’re trying to find in the trade market before February.”