Rangers administrator David Whitehouse says Bill Miller's offer for the club is acceptable at a commercial level and anticipates an improved bid from the Blue Knights.
The Blue Knights, in partnership with Sale Sharks owner Brian Kennedy, submitted a formal bid on Friday despite seeing major creditors Ticketus withdraw their financial support.
Administrators later revealed there was a significant difference in the value of the two bids on the table and Whitehouse has now confirmed only American Miller's £11.2million offer would have a chance of succeeding.
Miller's bid remains conditional on a favourable outcome over potential football sanctions but Whitehouse believes he is closing in on preferred bidder status if the Blue Knights do not increase their offer and try to satisfy the club's creditors.
Whitehouse said: "Both parties know the bid has got to be commercially at a level which would be acceptable to creditors.
"Bill Miller's bid is acceptable at that level. The only barrier to Bill Miller's bid is regulatory clearance.
"We are hopeful that, within a very short space of time, the assurances that Bill Miller needs in relation to his bid can be delivered by the relevant authorities.
"We would hope that, at the very early part of next week, Bill Miller's bid will be clarified whether he can proceed or not."
Whitehouse revealed the Blue Knights' bid was "very substantially lower" than the reported value of £13million and added: "We have been in ongoing dialogue with Brian Kennedy on behalf of the Blue Knights during the course of today and we understand that he may be revising his bid.
"We are encouraging them to re-visit that offer and get it at a level where commercially it may be acceptable to the creditors. If they are going to submit a bid at a level which would be commercially acceptable, they need to open very urgently a dialogue with the football authorities."
Rangers have appealed against a £160,000 fine and 12-month transfer embargo from the Scottish Football Association while the Scottish Premier League clubs meet on Monday to discuss stricter penalties for insolvency action.
Whitehouse insisted liquidation was not inevitable despite Rangers' liabilities being around £60million with a further bill of up to £75million possibly coming as they await the verdict of a tax tribunal.
"There is a very clear path which has been set to avoid liquidation and the regulators have been very clear in their desire to ensure that scenario doesn't prevail," Whitehouse said.
"Therefore, it's in the gift of the bidders to make that happen."
However, Whitehouse warned that time was running out - players revert to their original contracts on June 1st after accepting pay cuts of up to 75 per cent and the club will have no gate receipts for several months.
"I'm reluctant to quote deadlines because invariably they tend to get missed, but there is a very real and large looming deadline, which is the end of the season, which is the point at which we have funding in place to continue to operate the business," he said. "So, by definition, we have to conclude a transaction before that date and therefore that deadline is looming large.
"It's in the hands of the bidders to create a final date because they need to progress their positions to the point where they're capable of being accepted. I don't see any barriers in place to bidders being able to achieve that by the middle of this week."