As their ambitious project draws closer to reaching its end game, he has pleaded with the Edinburgh-based businesswoman to hang tough and retain her faith in administrator Bryan Jackson somehow managing to force over the line the one remaining deal that will take the club out of administration.
Jackson spent last week in the Lithuanian capital of Vilnius, convincing UBIG creditors to transfer their 50% shareholding in the club to the Budge-backed Foundation of Hearts and doing everything in his power to persuade lawyers representing UKIO Bankas of the worth of accepting the £2.5m on the table for their 29% stake.
He is waiting by the telephone for the call to return to the Baltic state with his BDO colleague Trevor Birch and attend their creditors' meeting, but, as of this moment, radio silence reigns from everyone involved with time ticking and First Minister Alex Salmond admitting he has contacted the Lithuanian ambassador.
Hopes remain that the gathering will take place, as originally suggested, this Friday. With BDO having already predicted that Hearts will run out of money at the end of this month and any deal with UKIO Bankas being subject to a 20-day cooling-off period, there is about as much 'wriggle room' left in this deal as the last plane out of Saigon.
Jefferies feels for everyone involved. The Dunfermline manager worked with Jackson and his staff as the club battled their way out of administration last year and understands - having been part of the Romanov regime at Tynecastle and witnessed the particular quirks of Lithuanian business practice - why he is finding the act of pinning down something as simple as the date of a meeting somewhat akin to nailing jelly to a wall.
The 63-year-old fears Budge will be running short of patience. He helped provide her with information on the infrastructure needed to run a football club when she first showed an interest in backing a supporter-led rescue bid, and has urged her to remain on board as Jackson bids to negotiate the final remaining obstacle in its path.
"I remember Bryan saying Dunfermline was a very, very difficult case, but I imagine this will have been the hardest of all because Hearts' problems lie thousands of miles away," said Jefferies.
"You can get frustrated with the apparent lack of progress, but I dealt with Lithuanians during my time at Hearts and it is just a different culture in business terms.
"They don't trust each other over there, so what chance have you got? That was just the feeling I always got, there was a certain coldness in your dealings with them and I think that was just the culture.
"I worked for them and it was hard, at times, to get much out of them. My concern is that the future of Hearts makes no difference to the lives of these people over there.
"It is just a name of a piece of paper and there will no doubt be other companies affected by what has happened at UKIO Bankas.
"Bryan is stuck in the middle and he has a tough, tough job in trying to get the club out of administration and into the right hands, but he has a good record in these matters.
"The Foundation of Hearts have come up with some great solutions and Ann Budge has put the money behind them, but how long will she remain patient? That is my great worry.
"I really do hope she sticks it out, but she's only human. I just hope she doesn't get disillusioned and I hope everyone can keep her on side and get the club through this.
"The fans have done fantastically well to keep the club going this long and I am sure they must feel they are flogging a dead horse at times because it must be really, really difficult to get information from Lithuania.
"There are good people involved in the Foundation, though, and I hope they get it over the line."
Salmond, a Hearts supporter, confirmed his involvement last night following a plea from BDO.
"At the bequest of the Hearts administrators, I phoned the Lithuanian ambassador to try and facilitate the important moves that are being made," he revealed.
"I should say, I'm a Hearts supporter and I declare an interest, but I have made the same approaches when necessary with other football clubs in Scotland who have been similarly struggling.
"There were helpful conversations and progress has been made." Should the club's future be secured in the coming days, Jefferies has also asked Budge and the Foundation of Hearts to remain committed to current team manager Gary Locke following a nightmarish season that has resulted in relegation to the SPFL Championship in the wake of the 15-point penalty that administration delivered.
"Everyone knows I am a big fan of Gary, but he was dealt a terrible hand with the points deduction and being left with a team of kids," said Jefferies. "He has been criticised and I find that distasteful because I maintain that there has been a big improvement in that Hearts team over recent weeks.
"Hearts have the potential to be a very good side in the Championship next season and Gary, I am sure, will have the last laugh. I just hope Ann sticks with the club and then sticks with Gary. It is when he receives proper support that he should be judged like every other manager."