According to Jared Weiss of USA Today's Celtics Wire, Boston is trying to stay "in position to strike" on Davis or "any other transcendent player that hits the market."
Weiss linked guard Marcus Smart's impending restricted free agency to Boston's hopes of landing Davis.
Every move Boston makes with financial implications is reportedly being viewed through the prism of how it will impact the team's ability to make a run at Davis.
If the Celtics sign Smart, Terry Rozier and Jaylen Brown in the coming years, it could adversely affect their ability to acquire Davis.
Davis is under contract through the 2020-21 season, and any trade for him would likely require the Celtics to give up big man Al Horford to match salaries, per Weiss.
If Boston wants to remain patient, it will have to traverse through three more seasons after this one before it has an opportunity to sign Davis in free agency should he opt to test the market.
The 24-year-old is once again putting up MVP-caliber numbers this season with 27.0 points, 10.6 rebounds, 2.5 assists and 2.1 blocks per game.
New Orleans' playoff hopes are in peril, though, with center DeMarcus Cousins out for the season. The Pels are currently seventh in the Western Conference and hold a 1.5-game lead over the ninth-place Los Angeles Clippers.
Davis is a five-time All-Star and two-time All-NBA selection, but he has only appeared in the playoffs once in his career.
With Boston currently tops in the Eastern Conference and built to remain in that spot for years to come, the Celtics figure to be an attractive landing spot for The Brow moving forward.