The paper said talk in the City was that Sir Peter was to step down and that the Sainsbury family - which owns 37% of the company - may be preparing to sell out for some £6bn.
Meanwhile, the Daily Telegraph said that a row has broken out between Sainsbury and the Takeover Panel over whether the supermarket chain should be included in a formal statement by entrepreneur John Lovering withdrawing from his Somerfield bid.
By naming Sainsbury in Lovering's statement, the supermarket group could be prevented from buying any Somerfield stores for at least six months.
The panel said Sainsbury must be included in the statement but Sir Peter Davis is strongly resisting. Sainsbury has insisted that it was not part of Lovering's bidding consortium, but it had agreed to buy 150 Somerfield convenience stores from it.