Allegri, 59, replaced Scudetto winner Antonio Conte at the Stadio Diego Armando Maradona in the summer.
Napoli have struggled under Allegri and currently sit in 10th in the Serie A, with just 15 points from their opening 11 games.
According to La Gazzetta dello Sport, Allegri has demanded the club sign a right-back, a physical midfielder, and two strikers in January to help turn things around.
It’s also expected that Noa Lang and Lorenzo Lucca, who were available in the summer, will be placed on the transfer list again.
The report names Shakhtar Donetsk right-back Vinicius Tobias, Sassuolo’s Darryl Bakola, and Monza hitman Gustavo Varela as potential options.
