The club's administrators, Duff & Phelps, have given the group, led by former Ibrox director Paul Murray, until Monday to submit a bid strong enough to take the club out of administration through a Company Voluntary Arrangement (CVA).
Failure to do so would leave only American businessman Bill Miller's £11.2 million bid on the table after Singapore-based Bill Ng yesterday announced he was withdrawing.
But experts last night warned that the complex deal proposed by Mr Miller would effectively liquidate Rangers and see a new company replace it.
Administrator Paul Clark last night confirmed that if the Blue Knights consortium – which has the backing of Ticketus – did not make a new offer, then Mr Miller would be given preferred bidder status.
A source close to the Blue Knights told The Herald: "The club is in serious danger. No matter how Bill Miller tries to dress up his plans, they will involve the club going into liquidation. This is something we are desperate to avoid as we wish the club to remain in its current form.
"The Blue Knights plan to meet with the administrators over the weekend for further talks and hope to be in a position to make a fresh offer by Monday. Hopefully, at that point the administrators will offer us preferred bidder status."
Mr Clark confirmed that further meetings with the Blue Knights were planned, saying last night: "I spoke to Paul Murray on two or three occasions today and have no problem with these discussions continuing over the weekend.
"If Paul is still interested to talk, we're interested to talk, but it now comes down to Monday. It has gone on long enough and we have to really bring it to a head once and for all."
The Blue Knights had originally hoped to make a fresh offer by last night and pay the non-refundable £500,000 exclusivity fee for the rights to be named preferred bidder. However, the group was reluctant to make an unconditional offer, as stipulated by the administrators, when Duff & Phelps could not guarantee the release of club owner Craig Whyte's shares.
Mr Clark, however, felt Whyte would not be "an impediment", adding that "one way or another these shares will be delivered".
Mr Miller's offer is conditional on Rangers being guaranteed a place in next season's Scottish Premier League, and them not being burdened by "further points deductions, fines or other punitive sanctions, relating to either the terms of my purchase or the actions of the prior administrations, beyond those levied during the 2011-12 season".
The American has so far refused to engage with either Whyte or Ticketus, believing the finance company that supplied Whyte with £27m with which to buy Rangers in exchange for future season ticket revenue would not accept any "pence on the pound" settlement.
The Blue Knights were last night back in discussions with Ticketus about a possible deal.
Duff & Phelps had originally hoped to have a new owner in place by mid-May but now accept that is unrealistic.
Mr Clark said: "We've always stressed that we would have liked to have our new bidder in place prior to the end of the season.
"I think that is now looking unlikely, but I think that there ought to be a new owner in place at the beginning of next season and so it will probably be more likely in the close season now."
Mr Clark also revealed that the club was in danger of running out of cash soon. He said: "Without season ticket sales, the money would run out towards the end of May or early June. So from that point of view we really do need to proceed with one of these bids.
"Mr Miller said to other potential bidders, 'Put up or shut up'; and we really are in that position that if no further and better bid comes along, we'll proceed with him next week."
A CVA would enable creditors to receive a percentage of the sums they are owed.
Asked whether he felt it now more likely that the club would exit administration via a CVA or through liquidation, Mr Clark said: "It's still in the balance between the two, but I must stress that if it's going to be a Blue Knights CVA then they really do need to proceed with all speed over the weekend."
Meanwhile, Mr Ng yesterday withdrew, saying he had become "frustrated with the process of dealing with Duff & Phelps, and the length of time that it has taken to reach this stage".
The administrators also revealed it has yet to receive any formal offer from Sales Sharks owner Brian Kennedy.
Mr Clark said: "We had a discussion with Brian and his advisers in the last few days and we made it clear to him the level of commitment that he would need to make, but he said he was not prepared to do that."