With negotiations to persuade Fraser Forster to sign a four-year contract still to be resolved, and back-up Dominic Cervi released, the only shot stopper on the club's books is Lukasz Zaluska.
Nine players have been shed by the SPL champions, but, as yet, no one has been brought in.
A central defender and a striker are high on the shopping list. But Brown is in no doubt what is the priority.
He said: "We need to get a goalie in the first place. We've only got one just now, Lukasz.
"I'm not sure what is happening with Fraser, but we hope he comes back.
"He's a great goalie and showed how well he can do in big games, espe-cially against Rangers."
Should Celtic fail to meet Forster's demands, Craig Gordon is one potential target waiting in the wings.
Asked if the free agent would be a good fit in the Celtic side, Brown responded: "I think he would be."
Brown has first-hand experience of how fit 29-year-old Gordon is after working with him when they were on Scotland duty in America last month.
"He's had a lot of injuries, but I've had a lot as well, and it shows you can come back stronger from them," said the man who played his best football in a Hoops shirt after returning from ankle surgery in December.
"Craig on the Scotland training ground? He's ridiculous. He seems to be everywhere. And, for a goalie to be everywhere is scary.
"It would be good (if he comes here) if we can't get Fraser. But we'll just have to see what happens."