Dundee director George Knight is adamant the club will still have a future if it goes into administration.
The First Division club must pay an Inland Revenue bill of £250,000, raising fears administrators may be appointed, as happened seven years ago.
Knight said: "I've been involved here since the dark days of 2003, and we've not come through what we've come through to see that undone."
The Dens Park directors say the club's future will be clearer within 48 hours.
They met at lunchtime on Monday and emerged to say that they are continuing to explore all avenues and are asking for patience from the fans.
Major investor Calum Melville recently tendered his resignation.
That led to negotiations between the club and the director, resulting in Melville saying he would remain on the board if still wanted and that he would continue to back the club financially.
A senior Dens Park source told BBC Scotland on Sunday: "HM Revenue & Customs (HMRC) are looking for the payment immediately.
"And they are being completely inflexible, with no room for negotiation at all."
Melville told the News of the World I feel we are being treated differently from other clubs and I would have thought HMRC has a duty to treat everyone in the same fashion.
"The words used were that they 'wanted to bring Dundee to account and bring the rest of Scottish football to heel'.
"Dundee FC are in a hole with HMRC. We've been running in arrears with them for 19 months and they are now demanding full payment for the outstanding amount, which is something around £250,000.
"We have offered £100,000, which has been rejected. I have also offered to pay the full amount in May next year and that has also been rejected.
"There is no question that the club is being used as an example and it has put us in a really difficult place.
"My feeling is that HMRC will write to us in the next week and tell us what their intentions are. There is a very distinct possibility of administration."
Meanwhile, Dundee's problems on the field of play deepened on Saturday with a 2-1 defeat away to Cowdenbeath.
Gordon Chisholm's side sit seventh in Division One, 10 points behind leaders Dunfermline Athletic.
These are desperate times for a proud club who were League Champions of Scotland in 1962 and reached the semi-final of the European club the following season.