Both men previously enjoyed a healthy relationship when Gattuso played under Ancelotti at AC Milan. Together, the pair won two Champions League titles, one Serie A crown and five other trophies.
However, the relationship between the two soured when Gattuso replaced Ancelotti as Napoli's manager in late 2019.
Ahead of the upcoming cup clash between Real Madrid and Valencia, Ancelotti was asked about his counterpart during the pre-match press conference.
The Italian said: "We had wonderful moments together, winning the Champions League twice, experiences I will remember forever. A rapport doesn't always continue in the best way.
"We had some problems that are purely personal and I don't want to talk about this."
He later added: "From the AC Milan team I coached there have been many coaches, each with his own style and knowledge. Gattuso, Inzaghi, Nesta..."
Gattuso was later questioned about Ancelotti's comments and confirmed their 'personal problems' stemmed from their time at Napoli. Nevertheless, the 45-year-old still has 'respect' for his old manager.
During his own press conference, the Valencia boss said: "When I went to coach Napoli at that time, the press and television talked about the team being good and there was a problem. But he knows I respect him a lot. It was a work problem.
"We are talking about one of the best coaches in the world. For my part personally, humanly and in football, I have a lot of respect for him."
He later added: "When as a player I saw that Carlo gave everything for the team, I was ready to throw myself into the fire for him.
"And when things weren't working in the Milan locker room, I spoke up for the good of the team. That was our relationship.
"Now the rapport is a bit… let's say it is no longer what it was. We haven't spoken in over a year. I respect him a great deal and have nothing to say. He's one of the best coaches in the world and we won a lot together."