The 30-year-old Israel international has one year remaining on a £36,000 per week salary that could see Pompey go into liquidation before the start of the coming season.
Birch has imposed a £5,000-a-week salary cap to help cash flow with the rest of the money being deferred.
But the debt remains and Balram Chainrai's offer to buy the club is on the condition that Portsmouth's highly-paid players are off the payroll at the League One club.
With no club likely to take on Ben Haim's hefty salary and little indication that he or his agent Pini Zahavi intend to give up the money, Birch claims that could mean liquidation within weeks.
"Ben Haim represents everything that has gone wrong at Portsmouth, the excessive wages that were given four or five months before they went into administration the first time," he told The Independent.
"There were no relegation clauses in his contract, and here we are in League One trying to negotiate a way out of it.
"The player and his agent haven't caused the problem but they are not helping.
"It's not just him, but his contract is by far the largest.
"The creditors have accepted Chainrai's offer but it is conditional on me reaching an agreement with those players. I'm working night and day to persuade them that we are running out of time."
Asked when the club will fold if an agreement is not reached, Birch added: "We are two or three weeks away.
"Then the players will get nothing anyway. It's brinkmanship, but the club is on a knife edge. Michael Appleton (Portsmouth's manager) can't sign any new players because there may not be a club."