Liverpool's owners George Gillett and Tom Hicks have 24 hours to accept an offer from Dubai International Capital to buy the club, BBC Sport understands.
Government-run Dubai investment fund DIC have been in talks to buy the Reds but have denied any time limit.
But BBC Radio 5 Live football correspondent Jonathan Legard said: "I'm told the offer from DIC is £400m.
"Mindful of how they were passed over 12 months ago with a previous offer, I understand DIC want an answer fast."
He added: "That means George Gillett and Tom Hicks have, according to one source close to the talks, 24 hours to take this offer or leave it.
"DIC's frustration with Liverpool's owners has become abundantly clear."
DIC confirmed on Tuesday it had been in talks with American duo Hicks and Gillett about purchasing a stake in the Anfield club.
"It's no secret that we have been in discussions with the current owners," said DIC chief executive Sameer al-Ansari.
"It's not easy because (they) are in dreamland about valuations."
Al-Ansari later clarified that one of the club's owners "had come out of dreamland", without giving further details.
"Of the two owners I'm told George Gillett in principal has agreed or is inclined to agree," said BBC sports editor Mihir Bose.
"Tom Hicks has not and the two partners have an agreement that one can buy out the other. That for DIC is a stumbling block.
"The fact that DIC have gone so public on this means they know that the offer is not being accepted and they are trying to use fan pressure, like the Americans did last year."
Texan billionaire Hicks said last month he was not planning to sell a stake in Liverpool, denying media speculation he was in talks with DIC over a sale.
He also revealed that Gillett could not sell his stake in the Premier League club without his approval.
However, both Hicks and Gillett have become increasingly unpopular among Reds fans, drawing criticism for their treatment of manager Rafael Benitez and their financial management of the club.
That has given fresh hope to DIC, who were considering buying Liverpool before the Americans joined forces to complete their own takeover in February 2007.
Hicks said in January he had talked to DIC "once" about a 10-15% stake, but DIC said the valuation was too high.
DIC are thought to want a controlling stake at Anfield, meaning they would need to convince Hicks to relinquish part of his 50% of the club as well as buying out Gillett.
Another obstacle is the refinancing package announced on 25 January that has loaded £105m of debt on to the club.
Of that, £60m is earmarked to kick-start the new stadium development at Stanley Park plus £45m for future player transfers and to meet the club's working capital needs.
DIC, which manages about £6.5bn of assets, is owned by Sheikh Mohammed bin Rashid al-Maktoum, the ruler of Dubai.
Al-Ansari was the man behind the takeover bid a year ago and looked to have succeeded until then Liverpool owner David Moores suddenly opted to sell to the Americans.
"It took me two weeks to get over that," said al-Ansari, who is a Liverpool fan. "But it didn't dent my passion - I still go to every match when I'm here."