The Anglo-Jamaican forward is viewed as a potential addition to the squad and could also provide support for Rasmus Hojlund, with the two forwards offering different characteristics.
The main obstacle could be Udinese's asking price. Davis is valued at more than €20m, while the club have previously made clear that they consider him difficult to replace.
Udinese sporting director Gianluca Nani has spoken highly of the 27-year-old and suggested there would be little incentive for the club to sell.
“I consider Davis one of the best in Europe in his role. It is difficult to find a replacement at his level. When he is fit, he is really unstoppable. Since his England days I have considered him a top player, but he has been hindered by injuries.”
Nani also stressed that Udinese have several attacking options but must remain mindful of their finances.
“Gueye and Bayo are high level but don't receive the right recognition, we also have many attacking midfielders and can play with three at the same time.”
“We have many offensive players. It is clear that when you analyse the market you can always improve, but we are Udinese, we have to be careful with the balance sheet so as not to create negative situations.”
Napoli will therefore have to decide whether Davis represents sufficient value to justify a significant January investment.