The Sainsbury family continued to block a private equity takeover of Britain’s third-largest supermarket chain last night, despite the bid consortium raising its offer by 20p to 582p a share.

The consortium tabled the offer yesterday with the board of J Sainsbury, which did not immediately accept or reject it but communicated it to family shareholders.

Lord Sainsbury of Turville and his family, who own 18 per cent of the company, are understood to be wary of allowing the company to be sold to private equity bidders and have continued to indicate that they would be unhappy with less than 600p a share. Other shareholders, including Robert Tchenguiz, the property entrepreneur, are also believed to want a 600p-plus offer.

The board is understood to be torn because it is minded to recommend the new offer, while other shareholders are thought to be willing to consider a sale at the new price.

The consortium of CVC, Blackstone and Texas Pacific Group has told the board that it will not launch a formal takeover bid unless Sir Philip Hampton, the chairman, is able to secure the support of the Sainsbury family.

Last week the consortium made a 562p-a-share offer for Sainsbury’s, but it was rejected by Sainsbury’s board after pressure from leading shareholders.

Yesterday’s offer represents a premium of almost 35 per cent to Sainsbury’s share price at the end of January, before bid speculation started.

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