The 20-year-old midfielder has been heavily linked with a move to the Premier League, but the Brazilian club remain determined 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," Kalil confirmed in an interview with Radio Itatiaia. "That's what I want.
"That was a discussion we have with him, that I want to keep Bernard at any cost. If there is an offer, we'll put into the negotiation table that it's important that he plays Libertadores."
Chelsea recently secured the signature of Wallace from Fluminense, adding to a growing contingent of Brazil stars plying their trade at Stamford Bridge, with Ramires, David Luiz, Oscar and Lucas Piazon all employed in west London.
Rafa Benitez has been appointed as interim manager until the end of the season, but it is not yet clear how much influence he will have over prospective signings, with Roman Abramovich likely to be the man orchestrating transfer activity behind the scenes.
Either way, Kalil accepts that he may eventually have to let go of Bernard - whether it's to Chelsea or otherwise - but is adamant that he'll only sell the Atletico star for the right price.
"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," he continued. "If you sell Bernard, how much will you spend to replace him? This is something I have to think about.
"There is nothing yet, only speculation, there have been 15 [rumours] every day. But I have been sitting here (as president) for four years. I know what is an offer and what isn't. And for now nothing serious has come."