Gattuso admits he will struggle to convince the Argentine to stay at the club and reports in Italy says they have already identified Krzysztof Piatek as his replacement.
Gazette Dello Sport claim the Genoa striker is their number one target as he has impressed with 13 goals in his first-ever season in Serie A.
Piatek’s form has attracted interest from all over Europe and Liverpool have been linked with a move for the Poland international, 23.
However, Higuain’s imminent departure from the San Siro could put paid to any hopes they have of luring him to Anfield.
Higuain is currently on loan from Juventus but is keen to cut his deal short so he can make the switch to Stamford Bridge to link up with Blues boss Maurizio Sarri.
The striker, 31, enjoyed his most prolific season under Sarri when the pair were at Napoli in 2015-16.
Higuain bagged 36 goals in 35 Serie A games and Sarri has admitted he would love the chance to work with him again.
And Gattuso suggested he might get his wish when he addressed the media following their 2-0 win at Sampdoria on Saturday.
“When a player makes up his mind it becomes hard to convince them otherwise,” he said.
“I have a relationship with him that features great honesty, we say things to one another’s face.
“At the moment, he’s our player and we’re holding on tightly to him. I don’t know what will happen.
“I’ve talked to him a great deal but it’s hard to give advice because the career of a player only lasts 13 to 14 years.
“It’s his mind, not mine. The most important thing is to talk as men, look each other in the eyes and speak the truth.
“I still don’t understand what Higuain’s unhappy about because I see a happy lad involved in the locker room.
“We’ll see what happens. If it was up to me, I’d keep him at my house, lock him up in a room and throw him some food every two to three hours”
Chelsea boss Sarri is desperate for a new striker to help misfiring duo Olivier Giroud and Alvaro Morata.
They have just six Premier League goals between them this season and Eden Hazard has been forced to lead the line as a result.
The Blues have also been linked with Bournemouth star Callum Wilson but might be put off by the Cherries’ £75million asking price.