Shares in Sainsbury's fell 4% on Monday after speculation that two more private equity firms have pulled out of the consortium seeking to buy the retailer.
The revival of Sainsbury's has made it an attractive bid target
According to reports, CVC is now the only consortium member still interested in the takeover after the withdrawal of Texas Pacific and Blackstone.
The consortium has reportedly increased its offer to 582 pence per share.
But the Sainsbury family, which owns 18% of the firm, is said to want an offer of at least 600p per share.
Neither Sainsbury's nor any of the private equity firms have commented.
A fourth company, Kohlberg Kravis Roberts, withdrew from the private equity group earlier in April.
A 582p per share offer would value Sainsbury's at about £10.1bn. But according to reports, the Sainsbury family is holding out for a higher offer.
The private equity consortium has until 13 April to make a formal bid under a Takeover Panel ruling.
Last week, Sainsbury's reportedly rejected a 562p per share.
The private equity consortium is said to be unwilling to go ahead with a formal bid until it can be sure of the support of the Sainsbury family.
Robert Clark, a senior partner at the analysts Retail Knowledge Bank, told the BBC that the latest offer put the deal "on a knife edge".
"The board agrees that 582p is about right," he said.
"However, you may come to the odd situation that they're not even prepared to put the books up [for examination] because the family won't agree."
He added that the consortium faced a problem in that a bid of more than 600p a share was "just over their threshold".
Recent sales figures have shown that the revival at Sainsbury's, which has been led by chief executive Justin King, is continuing.
Last week, the supermarket posted better-than-expected sales figures for the first three months of 2007, with like-for-like sales, excluding petrol, up 5.9%.
David Buik from international brokers BGC Partners told the BBC that the recent sales recovery was another reason why the Sainsbury family was reluctant to accept a bid.
"I think they're also worried that they're half-way through this three-year rationalisation plan and I think they feel they're selling it before they should," Mr Buik said.
"I think also they remember in the back of their minds the problems when Morrisons took over Safeway - how the regulatory authorities took 11 months to decide that - of course [in that situation], morale goes down and the company is in danger of getting trashed."