The Israeli international has refused to accept a reduced pay settlement to walk away from the troubled club.
Birch told ESPN: "I am trying to fight to save the club going into oblivion but we only have a few days left and it's on a knife edge, I cannot genuinely tell you what will happen.
"Ben Haim, though, has a deal on the table but he has taken a view and doesn't appear to be shifting. He might think it's a game of bluff and the club won't close, but I think he will very soon find out that it is real.
"It remains still only 50-50 that the club can survive. We have until August 10, but we will all need to know a few days before Friday."
A team of Pompey trialists produced a superb performance to win against Bolton Wanderers at Fratton Park on Saturday at Fratton Park.