The private equity group seeking to buy Sainsbury's has increased its proposed takeover offer to 582 pence per share, according to press reports.
The revival of Sainsbury's has made it an attractive bid target
However, the Sainsbury family - which owns about 18% of the company - is reportedly still holding out for an offer of at least 600p per share.
The new deal would value the retailer and supermarket chain at about £10.1bn.
The private equity consortium has until 13 April to make a formal bid under a Takeover Panel ruling.
Last week, the consortium - which includes private equity firms CVC, Blackstone and Texas Pacific - lost one of its members when Kohlberg Kravis Roberts (KKR) pulled out.
Last week, Sainsbury's reportedly rejected a 562p-a-share indicative offer. The company declined to comment on the reports of an increased offer.
Reports suggested that a higher bid approach could lead to a split among the supermarket's major shareholders.
The Financial Times reported that the Sainsbury's board had indicated to the bidding consortium that the new offer was high enough, but that the sticking point remained the Sainsbury family.
The private equity consortium is reportedly unwilling to go ahead with a formal bid until it can be sure of the support of the Sainsbury family.
Lord Sainsbury of Turville and his family own an 18% stake in the supermarket chain, and other major investors, including property entrepreneur Robert Tchenguiz, who has a 5% stake, are also said to be holding out for a higher offer.
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 the 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."