According to reports from Italy, Calzona has agreed to stay in command of Slovakia till the completion of the current campaign.
Mazzarri was commended by owner Aurelio de Laurentiis for “having helped the team in a tricky period”.
On Wednesday, Napoli will play Barcelona in their Champions League first-leg match.
After taking over as Slovakia’s manager in August 2022, Calzona assisted the country in making it to the European Championship in Germany this summer.
The 55-year-old has served as Maurizio Sarri and Luciano Spalletti’s assistant at Napoli in the past.