Bryan Jackson, the club administrator, spent yesterday morning convincing creditors of major shareholder UBIG to transfer their 50% holding to Ann Budge's BIDCO 1874 group at a meeting in the Lithuanian capital city of Vilnius and is expected to remain there indefinitely to continue negotiations with lawyers representing UKIO Bankas.
UKIO possess 29% of the shareholding, along with a charge on Tynecastle Stadium, and there has been no deal struck with them as yet despite several hours of talks with Jackson and his BDO colleague Trevor Birch.
Budge has committed £2.5m to the Company Voluntary Arrangement that would take Hearts out of administration and plans, in time, to pass full control over to the supporters via Murray's organisation.
Jackson has warned that the Edinburgh club will run out of money by the end of this month and one insider last night told Herald Sport he believes the battle for its future will "go to the wire" as a high-stakes game of brinksmanship continues in the Balkan state.
However, Murray has been encouraged by the information he has been receiving and is daring to dream of a positive outcome as quickly as the end of next week. "We had two final major hurdles to get over," he said. "We are probably over one hurdle and we've got one leg over the other.
"I am absolutely delighted the UBIG creditors' meeting approved the transaction, which is a fantastic step forward, and slightly disappointed the UKIO Bankas creditors' meeting did not take place.
"However, I understand BDO and UKIO's lawyers had a lengthy, productive and positive meeting and we hope we can now get a creditors' meeting in place quickly and get this final hurdle cleared by the end of next week.
"If I can use a footballing analogy, we would have taken this result before the game. It wouldn't be Hearts if it wasn't being done at the 11th hour, but we are significantly further forward than we were."
Despite the positive noises being made last night, UKIO have been difficult to pin down on their intentions with reports suggesting they had been unsettled by talk of counter offers to the one put forward by Budge.
Bob Jamieson re-emerged over the weekend to claim he would return to the table should Budge's proposal fail to go through while sources in Lithuania spoke of a late offer being submitted by email, with Pat Munro, linked with previous takeover bids, reported in some quarters to be the architect.
"Anything that muddies the waters at this moment in time is unhelpful," said Murray. "BDO have spent a considerable number of months trying to clear those waters and it is very difficult to try to deal with something that comes in at the 11th hour and has not been through any due process.
"Everyone who has been involved in this process has the best interests of the football club at Hearts, but these are incredibly delicate discussions taking place.
"I don't want to keep using footballing analogies, but we are at the point where the referee has the whistle in his mouth and is looking at his watch. The last thing you want is a complication to waste any more time when you are trying to get the ball down the park."
There were suggestions ahead of yesterday's meetings that Budge would walk away should the transfer of shares fail to take place, but she has been kept fully informed of all developments and remains most definitely on board.
"Ann is fully committed to the process and has been for over three years, but she is as frustrated as everyone else that it has taken so long," said Murray.
"BDO have been clear about the fact the club will run out of money by the end of April and this has to be done by then."