"I'd hope to know [their answer] by the end of the week," he said.
But yesterday City Care reiterated to The Star its position of last week - that they will not be granting any extra time off to their employees.
Said Mr Chappell: "The next move will be to approach CCHL if it's a flat 'No', which I suspect it might be."
But CCHL chief executive Bob Lineham was not offering the union any hope.
Mr Lineham said the company "will stay out of it entirely".
"Part of our philosophy is that we do not get involved in the management of individual companies," he said.
"It's up to them how they manage their staff."
Said Mr Lineham: "Our response [to the union] will be that we do not interfere with the management of City Care."
Mr Lineham would not comment on whether he thought City Care workers should receive extra leave along with city council staff.