That led to negotiations between the club and the director, resulting in Aberdeen-based Melville saying he would remain on the board if still wanted and that he would continue to back the club financially.
It is believed that the club are trying to raise the money to satisfy HM Revenue and Customs (HMRC) and that a leading figure in the insolvency world is currently negotiating with them on the club's behalf.
But room for manoeuvre is limited. 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."
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.
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