Cavani is a free agent after leaving Old Trafford at the end of the 2021/2022 season.
He had been linked with a move to Nice in France but wanted to test himself in Spain and that move is on the brink of coming to fruition.
The 35-year-old joined the Red Devils from Paris Saint Germain in the summer of 2020 and proved to be a revelation in his first season, scoring 17 goals in 39 appearances across all competitions as Manchester United finished in second position in the league.
The 2021/2022 campaign proved to be a challenging one for Cavani on a personal front and the club in general, as the Uruguayan struggled with injuries and scored only two goals in 20 appearances while United finished sixth on the table.
Speaking ahead of Monday's clash with Atletico Madrid in LaLiga, Gattuso said: "I am optimistic by nature, we are working, talking, I think we have possibilities.
"[Valencia owner] Peter Lim is working hard, and I think we are going to sign Cavani because Lim, at the moment, believes a lot in this possibility.
"When you talk about Cavani, he is not a normal player, he has played for big clubs and at the highest level for a long time. He is 35 years old but has an incredible mentality, I am talking about the technical part, not money, that is not a problem, there are two different things.
"The problem is that a lot of people talk in an operation and it's not easy to close it in two minutes.
"We are working on it for two or three weeks, but we are going to wait and I hope that it will close in a positive way for us. We have done everything to solve it and I hope that the signing will be closed.
"We are talking about a champion, a footballer who has played in teams of a high level, I am also excited that he comes here."
Valencia have won one and lost one of their two La Liga games so far.
