The 26-year-old scored in the first half of Uruguay’s 2-1 loss to Brazil in the semi-final of the Confederations Cup on Wednesday night.
Earlier this week, De Laurentiis urged the striker to make a decision on his future by 19 July, and Cavani wants to enter discussions with his current employers.
“I love Naples. It’s my town and my heart is there. But when I was here in Brazil I’ve heard untrue rumours and things about me,” said Cavani.
“For this reason, now I want to talk as soon as possible with president De Laurentiis.
“We must clear ourselves. I’ll talk with him, as it’s done between men, really soon.”
Cavani has regularly spoken of his admiration for Real Madrid and has also expressed an interest in working with Chelsea boss José Mourinho.
But Napoli’s new manager Rafael Benítez has urged the striker to remain at the Serie A runners-up for another season and ignore interest from his suitors.
“Neither me or the president want Cavani to leave because he’s a really important player for us,” Benítez told Radio Marca.
“Clearly, there is a clause that you can’t control, but we want him to stay to score many goals and to continue growing with Napoli.
“He found success and fame with this team and he can stay for another year with supporters who love him. Napoli don’t need the money. Should he leave, it will be for his clause, and it’s a really high sum.
“But our intention is that he will stay. The president’s intention is to strengthen the team and Cavani is our most important player.I don’t know if he has had any offers, but I know there is a clause and anyone who wants him will have to pay it.”
Cavani netted 36 goals in 41 appearances for Napoli last season.