Hearts supporters are closing in on the 3,000-mark of extra season ticket sales, which they were challenged to reach in a two-week period ending on Friday.
Supporters also raised £30,000 at a rally on Sunday and the Foundation of Hearts group has secured direct debits from most of the 6,000 fans who pledged money towards a takeover bid.
Speaking of the fans' response, Birch said: "It's been tremendous and we are very close to the 3,000 target that we set ourselves.
"We are just over 2,800 now so I would have thought by the end of the week we'll be over that target, so it has been a fantastic response."
Birch confirmed that the "three or four" interested parties who emerged at the start of the process were still in the frame to take over with the deadline for first offers set for July 12.
"How that progresses, you can't really say," he added.
But he feels the Foundation of Hearts group is a credible bidder and offered encouragement to their efforts.
"I think in any period where you have had ownership that effectively takes the club out of the community, if it then becomes owned by the fans and comes back to the community, it almost feels like it has been cleansed, and that can only be a good thing," he said.
Birch added: "They are as well able to bolt on to whoever else might be interested as well, if they don't buy it in their own right."
On the upcoming deadline, Birch said: "It's only the first stage. We are trying to eke out whether there is any real interest in there and then work with those parties who do make that bid to then move towards a preferred bidder."
Birch and BDO colleague Bryan Jackson inherited a "desperate" situation when they took over a club that had run out of money and had no guaranteed source of immediate income last month.
Thirteen staff were made redundant but Birch confirmed no players would follow them out of Tynecastle after several agreed wage cuts and John Sutton moved to Motherwell.
Hearts have now plugged the funding gap which will allow them to continue with their current squad for the coming months.
"It's a lot better now," Birch said. "We literally had no cash and we probably have about £750,000 now from all these activities, so it should give us that breathing space.
"I'm confident and hopefully we can resolve the ownership issues.
"But I'm confident we have the wherewithal to go forward in the short term, which will give some of these interested parties time to frame their offer."
In the meantime, Hearts fans have been urged to keep supporting the club financially by buying more season tickets, match tickets and hospitality, as well as donations.
BDO wants to ensure there is enough money to keep funding the club until the end of the year as they bid to resolve outstanding issues in Lithuania.
The firm is liaising with lawyers to ensure shares owned by administration-hit Ukio Bankas and UBIG are included in any sale. The bank holds security over Tynecastle and both firms are owed a total of £25million by the club, thus holding the key to any Company Voluntary Arrangement.
Birch said: "We are only dealing with the legal team to the (Ukio Bankas) administrators at the moment.
"It's a bit difficult with UBIG because although technically they are in an insolvency procedure, they haven't had an administrator appointed yet."