Hamsik scored what proved to be the match-winner as Slovakia defeated Russia 2-1 in their second Euro 2016 Group B outing on Wednesday.
The 28-year-old also set up the opening goal for Vladimir Weiss, with Slovakia edging closer to progressing into the knockout stages.
Hamsik has been playing at Napoli since 2007 following a €5.5million move from Serie B outfit Brescia, but Kozak says it is time he found a new club.
"He deserves a bigger club for his performances," Kozak said.
"I think Napoli has become too small for him. He deserves to play at a really big club."
He added: "He was superb with the ball but also helped a lot in defence and scored a beautiful goal.
"To that point, it was all so tight and so tense, there was no space in which to play for either side.
"We were only going to break the ice through an individual bit of genius and we got that from Hamsik and Weiss. They were fantastic."