However, that figure was way short of Napoli's valuation of the Uruguayan, who did move later in the summer to Paris Saint-Germain for £53million.
"Cavani, Manchester City visited me in Rome," explained De Laurentiis as he talked exclusively on Sky Sports News.
"But they just offered, I don't remember exactly but around 35 million [euros] so I didn't accept - there was a high level of interest."
De Laurentis is in England as Napoli face Arsenal on Tuesday evening - a game you can see live on Sky Sports 2HD.
"It is very tough as the Gunners are very strong, we met them last August but it will be a different game. We play without fear so I hope it is a good game," said the Napoli chief.
De Laurentis is delighted with his new coach Rafa Benitez, who he says he did not even think about appointing until after the Italian championship had finished and the departure of Walter Mazzarri - now Inter Milan coach - was confirmed.
"I respect my work, until the Italian championship ended, I decided we would not be in touch with anyone else and wait for Mazzarri to say no, the day after I flew to visit Rafa Benitez," he revealed.
"He convinced me immediately. We stayed eight hours in a London hotel and everything was defined.
"Rafa is a very nice man, he is always smiling and is relaxed and I hope it will be a very long relationship."
The Napoli owner, who is a media mogul away from football, also revealed he was able to win the race to sign Gonzalo Higuain by simply going straight to Real Madrid.
"It was very fast as we were negotiating with a Brazilian guy, now all the soccer players have to belong to me and their image rights, for me it doesn't work without it - and this Brazilian guy had a lot of trouble," he said.
"Then I said to my guy why don't we go to Real Madrid and in 10 minutes it was done. Then with Higuain his agent said fine and the day after in Venice, closed in a room for eight hours we did a deal.
"I didn't know about Arsenal were interested! Sorry Arsenal," he joked.
De Laurentiis confirmed that he was looking at investing in an English club, in a similar way to which Udinese owners the Pozzo family bought Watford a few years ago.
"I am very attracted by England and United States," he said.
"The United States because it is new land and you can make a club from zero and England as it is the real soccer domain.
"I am interested to explore but right now I am just exploring as I am busy with the movie industry."