The consortium bidding for British supermarket Sainsbury's has reportedly upped its offer to 582p per share.
But the Sainsbury family, which owns almost a fifth of the group, is said to be holding out for a 600p per share bid.
Last week the consortium, which includes CVD Capital, Blackstone and Texas Pacific, reportedly had an offer of 562p per share knocked back by Sainsbury's.
If true, the latest reports of a renewed bid would value the supermarket chain, the UK's third-largest, at £10.1 billion.
Under stock exchange takeover panel rules, the consortium must make a formal bid by this Friday.
Any prospective takeover was complicated last week by the consortium losing partner Kohlberg Kravis Roberts (KKR) after the private equity group withdrew.
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