The offer from the consortium, led by private equity firm CVC Capital Partners, follows a 562p a share offer which was reportedly rejected by Sainsbury's last week.
However, the revised offer is likely to be rejected today because Sainsbury's will not consider an offer below 600p a share, according to reports in the Financial Times.
The consortium has until Friday to make an offer, after being given a 'put up or shut up' deadline last month by the Takeover Panel.
“The sticking point is the [Sainsburys] family,” a source told the Daily Telegraph. “If the board can deliver the family we have got a deal. But we cannot do this without the support of the family.”