The New Zealand forward has been trialing with the Dons in recent weeks but there is plenty of interest from the likes of Doncaster Rover and Bristol Rovers.
Brown admits there is an offer on the table for the ex-Plymouth Argyle man but says he will not push into a decision.
"I'm still waiting for a response from him directly or indirectly through his agent. That would come to chief executive Duncan Fraser, who is handling the situation now," he told the club's official website.
"I'm putting no pressure on Rory Fallon or Duncan Fraser because I don't want to be unfair and ask for a player who may well be outwith our budget.
"It's not that he's asking for an exorbitant salary, it's just that we have reached the player budget as far as I understand so I need to be fair and let Duncan deal with things from now on."
As for reports suggesting Fallon was holding out for money, Brown disagreed with that speculation, saying there is actual interest from other parties.
"I don't think this is a major money issue, this is an issue with regards to the player's feeling and the competition we undoubtedly have for him," he added.
"I know from speaking to his agent - and he is an honest agent - that there are other clubs expressing an interest in him.
"We've got to be patient and I would generally rather wait and let him come back to us, rather than keep chasing him.
"I have chased him, of course. When he was here, I spoke to him very seriously about coming here and he seemed well-disposed.
"He is friendly with one or two of the players already and he liked the place. But he's got a lifestyle change to consider and I think that's what is delaying things."