The three players have been linked with a move away from Naples this summer.
Hamsik is a reported target of AC Milan while Lavezzi and Cavani have attracted the attention of Premier League clubs.
"From what I know, all three will stay at the club," said Mazzarri to Gazzetta dello Sport.
"They are all willing to start the season with the Napoli colours."
Napoli finished third in Serie A last term and will play in the Champions League next season.
Source: Team Talk
Source: Team Talk