Rhone became the first suitor to show their hand when the New York-based fund management firm, run by billionaires Robert Agostinelli and Steven Langman, presented their offer to Liverpool in the early hours of March 13.
It is believed they informed the club they expected to discover whether their bid had been successful by April 5.
There has been no further contact between the parties and Telegraph Sport understands that Rhone are not prepared to extend that deadline.
Though it is believed Rhone's offer met Liverpool's valuation, it is thought the level of control they hoped to acquire for their stake as well as the nature of their investment has proved a stumbling block. Hicks is believed to be particularly resistant to seeing his stake being decreased.
Rafael Benítez, the Liverpool manager, had met with representatives of the group to discuss their plans for the club and it is believed he had kept senior players, including Steven Gerrard and Fernando Torres – who has been unequivocal in his demands for Liverpool's owners to back Benítez in the transfer market this summer – abridged of developments.
That Rhone's deadline – barring an unexpected and unlikely turnaround in the next 24 hours – will pass with no progress made will no doubt come as a blow to those attracted by the group's promise of a £25 million infusion of funds for transfers, but it is far from the only cause for concern for a club whose immediate future remains clouded.
On the pitch, Liverpool know they must beat Birmingham at St Andrews on Sunday to maintain the pressure on Tottenham Hotspur and Manchester City in the race to qualify for next season's Champions League. Off it, sadly, matters are far more complicated.
The Royal Bank of Scotland, Liverpool's bankers, have informed Hicks and Gillett that they must reduce their £237 million debt burden by £100 million by July if they are to be granted a deal to refinance their loans.
Christian Purslow, Liverpool's managing director and the man charged with securing a cash infusion to reduce the debt, has consistently identified Easter as the time by which he "hoped" to have a deal agreed with an outside investor to bring an end to the stagnancy induced at Anfield by the current, unpopular regime and to enable work to begin again on the long-awaited new stadium on Stanley Park.
Though sources at the club insist that was more guideline than deadline, that the holiday period will pass with Liverpool no closer to securing their financial future will hardly inspire confidence for Benítez's squad or the club's fans.
Sources at Anfield, though, remain confident of attracting the required investment "in good time" for the club's loans to be refinanced.
As many as "six or seven" serious investors were believed to be looking at matching or bettering Rhone's offer three weeks ago and noises emanating from the club suggest that as many as two of those are expected to materialise into firm proposals.
What is not in doubt, though, is that Liverpool are reaching their end game.
In the next three months, Hicks and Gillett's tenure will almost certainly reach an end, at least in its present form. More straightforwardly, defeat at Birminghamon Sunday and against Benfica in the Europa League second leg on Thursday, and much the same may be said of Benítez.
