The Blues are thought to be lining up a bid for the 20-year-old midfielder having recently secured the signing of fellow Brazilian Wallace from Fluminense.
But Kalil does not want to let go of the exciting prospect and is keen to keep him for the Copa Libertadores, which ends next summer.
"I don't want to sell Bernard, if I have to lose money to keep him for Libertadores, I will. That's what I want. That was a discussion we have with him. that I want to keep Bernard at any cost," he told Radio Itatiaia.
He added: "If there is an offer, we'll put into the negotiation table that it's important that he plays Libertadores."
Kalil does accept that he may eventually have to let go of Bernard but is adamant that he will only sell him for the right price.
He added: "This is football. I'm not afraid of selling anybody. But I won't give anyone for free. I have to sell and sell well.
"If you sell Bernard, how much will you spend to replace him? This is something I have to think about."
Kalil says no formal offer has yet been made despite the speculation linking Bernard with a move to the Premier League.
"There is nothing yet, only speculation, there have been 15 [rumours] every day. But I have been sitting here (as president) for four years," Kalil said.
"I know what is an offer and what isn't. And for now nothing serious has come."
Don't miss any live and exclusive football coverage this festive season. Click here to visit Sky Sports' Festive Calendar