According to ESPN.com's Brian Windhorst, teams are preparing to strike if Davis does not accept an extension from the Pelicans.
The forward can become a free agent in 2020 by opting out of the final year of his deal.
During the early part of the 2018-19 season, the Pels are among the NBA's best teams at 3-0.
Also, Davis is perhaps the leading MVP candidate with per-game averages of 30.3 points, 13.0 rebounds, 5.3 assists, 3.3 blocks and 2.0 steals.
The 25-year-old has been named an All-Star in each of the past five seasons, and there is little doubt that he is the NBA's premier big man.
Davis is in line for a massive contract regardless of his decision, but he can make the most money by re-signing with New Orleans on a supermax deal.
ESPN's Adrian Wojnarowski appeared on the Brian Windhorst & The Hoop Collective Show (h/t RealGM) this week and relayed his feeling that Davis is looking for reasons to stay with the Pelicans:
"My sense has always been with Anthony that he is always: some players are looking for reasons to stay and some players are looking for reasons to leave. I've always thought that Anthony is a guy looking for reasons to stay in New Orleans. And I think that will continue to be the case until he feels he's out of them."
While the Pels haven't advanced past the second round of the playoffs during Davis' tenure, they appear to be turning a corner.
As Nikola Mirotic, Julius Randle, E'Twaun Moore, Jrue Holiday and Elfrid Payton put up impressive numbers in support of Davis, New Orleans could be a threat to Western Conference stalwarts such as the Golden State Warriors and Houston Rockets.
Davis is a franchise-changing talent who will likely be the apple of every general manager's eye, but there is still risk in trading for him. If he doesn't accept a contract extension from New Orleans, any team that trades for him would be doing so with the knowledge that he could leave after one season.
That was the case when the Oklahoma City Thunder traded for Paul George last year, but the gamble paid off when he re-signed in July.
Since the potential reward is so huge, the Pelicans shouldn't have any issue finding a trade partner if they can't work out a long-term deal.