The leader of City of York Council, Dave Merrett, says despite the best efforts of all concerned it is difficult to find someone who will take the club, and its problems, on.
The Bootham Crescent team has until Saturday to put a deal in place to secure its future, or the home game against Swansea could be the last played by the club.
The Minstermen's difficulty lies not only in the significant financial hole, but also the uncertainty over the club's future at the Bootham Crescent ground.
The city centre ground, owned by a separate company called Bootham Crescent Holdings which includes ex-city chairman Douglas Craig among its directors, has been the subject of interest from housing development companies.
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I have to say it is not looking good, the reality is there is a significant hole in the football club's funds
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Leader of City of York Council, Dave Merrett
Councillor Merrett told BBC Radio York:"We (the Council) have all along tried to facilitate all the interested parties, anybody interested in saving the club.
"I have to say it is not looking good, the reality is there is a significant hole in the football club's funds and actually trying to find somebody who is willing to contribute to sort out the short term is the key issue, and it is difficult."
No-one from the club's administrators Jacksons Jolliffe Cork was available for comment.