That is according to ESPN NBA expert Adrian Wojnarowski.
On Monday, Davis shocked the NBA world when his agent Rich Paul told Pelicans officials that the five-time All-Star wanted to be traded.
And a move to the Lakers was cited as his most likely next destination.
The Pelicans can offer Davis a supermax deal worth $240million in the summer.
But the 25-year-old wants to leave New Orleans and join a championship contender.
The Pelicans had been resistant to any trade talks with the Lakers.
But Wojnarowski has revealed executives from both franchises have held talks over the phone about Davis.
He wrote on Twitter: “Lakers and Pelicans executives have connected on the phone concerning Anthony Davis, league sources tell ESPN.
“They’re expected to talk again prior to next week’s trade deadline.”
The NBA trade deadline closes on February 7 so the Lakers face a race against time to convince the Pelicans.
The Pelicans are currently 13th in the Western Conference and have lost their four of their last five games.
This season, Davis is averaging 29.3 points, 13.3 rebounds and 2.6 blocks for the Pelicans.
Davis has missed the Pelicans last six matches with a finger injury and head coach Alvin Gentry said he wasn’t sure if the centre had played his last ever match for them.
“That’s a hard thing to answer,” Gentry said.
“I don’t really know how to answer that.
“I assume that he will [play for us again] but that’s something that needs discussing and seeing what’s best for him and what’s best for our team.”
ESPN pundit Stephen A. Smith believes LeBron James needs the Lakers to complete a trade for Davis otherwise he will never win an NBA title again.
“The Lakers, unlike the Paul George sweepstakes, the Lakers don’t have the same kind of leverage now,” he said on the Stephen A. Smith show.
“Here’s where the problem lies. The Pelicans don’t have his rights until the end of this season, they have it until the end of the following season.
“The Pelicans can sit there and stand pat and simply say, ‘we’re not moving him anywhere until this summer’.
“Well by the time this summer rolls around, Boston’s a competitor and Anthony might consider going to Boston. He might be willing to commit to a long-term deal with some of the talent they’d still have in place to play alongside with him.
“If you’re the Los Angeles Lakers, whereas you played hard ball and you wanted to hold on to a Kyle Kuzma because you didn’t want to let him go, that ain’t necessarily the case now. You don’t have that kind of leverage now.
“If you’ve got to give him up to get Anthony Davis, that’s what you’ve got to do.
“If you’re the Lakers, you literally don’t have that choice at this point. You’ve got to be willing to give the New Orleans Pelicans anything they want outside of LeBron James, you have to do that now. There’s no way around it.”