The Foundation of Hearts is one of three groups hoping to purchase the administration-hit club and save it from extinction.
Reports on Wednesday, however, have suggested rival bidder HMFC Ltd has been told the company backing it - American firm Club 9 Sports - has withdrawn £5million of funding.
Meanwhile, the other group involved in the running, Five Stars Football Ltd, lists controversial former Livingston owner Angelo Massone among its directors.
But in an open letter supporters backing FoH, the group's independent chairman and Labour MP Ian Murray insists the fan-backed bid offers the best hope of survival.
Murray states: "The media have reported that HMFC Ltd and Five Stars Football Ltd are the other two bidding parties. The Foundation at this stage will not comment on the bids of other organisations.
"There's been a fair amount of comment and conjecture, and perhaps even a bit of scaremongering from one or two individuals, but we've said all along we're going to focus on what matters to us: Heart of Midlothian.
"The administrators need a period of time to do their job and we will just let them get on with that job.
"The Foundation of Hearts is confident that the bid submitted on its behalf Is the best financial bid for Hearts and its creditors; has cash ready to complete the transaction immediately; provides the strongest possible working capital solution; delivers lowest risk for creditors and the future protection and growth of the club; combines the best business people, and their capital, with genuine supporter interest and influence; and delivers a highly professionally run club with a clear way to engage Foundation members to the benefit and growth of the club."
Hearts owe £25million to Lithuanian firms Ukio Bankas and UBIG, who are both insolvent, although the latter has yet to appoint an administrator. UBIG owns 50 per cent of the club while Ukio Bankas holds a 29.9 per cent stake and a floating charge on Tynecastle as security on the debt.
But FoH is confident it can get the club back on an even keel.
It has pledges from 6,000 supporters vowing to make cash contributions towards that will help pay for both the purchase of the club and operating costs.
Administrators BDO are now running Hearts and hope to achieve an exit from administration via a company voluntary arrangement (CVA), but that can only be achieved if those due 75 per cent or more of the total debt accept a pence-in-the-pound offer.