That is according to USA Today, who claim that the supermax contract he will be offered will decide his fate.
The 25-year-old can be offered a deal worth $239.5million (£186m) in the summer.
Davis is rumoured to be closest to a trade move to the Los Angeles Lakers, as the franchise is looking for a co-star for LeBron James.
However, they will be challenged by the Golden State Warriors and the Boston Celtics, who are both interested in acquiring the star.
The Warriors reportedly want Davis to be their future leader of the franchise, in the event of Kevin Durant and Klay Thompson leaving this summer.
Both are expected to become free agents in the summer, and the Warriors will need to sign a replacement.
The Warriors could be in the best situation to land Davis if the report from USA Today is true.
Frank Urbina wrote that the Pelicans will hang on to Davis past the trade deadline of February 7.
“No matter what happens over the coming weeks, there’s little chance New Orleans deals Davis by February’s deadline,” he wrote.
“If something were to come to fruition, it’ll happen this offseason, when the Pelicans will have their first chance to offer Davis a monster extension.
“It’ll depend on what Davis does with the extension offer.
“If he turns it down, he will almost certainly get traded in the summer of 2019.
“If he accepts, he’s not going anywhere for a very long time.”
Pelicans owner Gayle Benson admitted that it was possible that Davis would leave, as she insisted she wouldn’t stand in his way.
She told The Athletic that, while she was a big admirer of the star, if he wanted to leave, they would allow him to.
Davis, who has remained quiet on his future, caused controversy on Friday as he ‘liked’ a comment on Instagram suggesting he would snub a move to the Lakers.