Napoli appointed Italian Francesco Calzona to be the football team's new coach after sacking Mazzarri.
Calzona, who is the current manager of Slovakia, worked as an assistant manager with two of the club's former coaches Maurizio Sarri and Luciano Spalletti.
It is expected that Calzona will continue his role and lead them at Euro 2024.
The contract of the Italian coach with the club is until the end of the season.
But, his contract with Slovakia is until 2025, and could be automatically extended if they qualify for the 2026 World Cup.
"The Slovak Football Association is aware this is a non-standard decision that may cause concern among the public," the Slovak Football Association said.
"However, it also understands the exceptional relationship of Calzona to the club where he worked for a long time and wants to help him in the current difficult situation.
"Calzona has repeatedly declared that the main priority of his activity remains the Slovak national football team and its preparation for Euro 2024," it concluded.