With Napoli winless in eight matches across all competitions, reports in Italy claim the struggling Serie A side has decided to dismiss Ancelotti after the Champions League fixture against Genk on 10 December.
Rumours of a dressing-room revolt against Ancelotti have emerged, with former Milan boss Gattuso lined up as his potential replacement, while ex-Juventus coach Massimiliano Allegri and Luciano Spalletti have also been touted.
Outspoken owner De Laurentiis rejected reports Ancelotti is on the brink.
"My faith in Carlo Ancelotti has never failed," De Laurentiis told Radio Kiss Kiss. "My trust in Ancelotti is total and unconditional, as a man and as a coach.
"For the future I have decided nothing, it would be a mistake to decide at this moment, we have a season in progress, there is only one way: we must find peace, the team must return to work with serenity.
"I didn't call anyone, I never called Gattuso and I never heard Spalletti. With Allegri we talk, but not from today. We have been talking since 2013, we have a cordial relationship. In 2013 I called him because I wanted him to become the coach of Naples, at the time he was the coach of Milan. He told me that we would be ready at the end of the season. He called me and gave me his total availability but I am a man of honour and I went to London, I shook hands with Rafa Benitez, so nothing was done.
"I want to reiterate that my faith in Ancelotti is total, then we could have problems. The problems, perhaps, are in the work methodology, but this methodology allowed Carlo to win everything. He won a lot, he won everywhere. The top players he had, such as Cristiano Ronaldo, they also trained in a personalised way.
"It seems to us that there is a school system, so we must dictate the rule of the methodology. Our coach is Ancelotti, let's continue like this."
Napoli – eight points adrift of the fourth and final Champions League place – travels to Udinese this week.