Texas pacific, CVC and Permira are part of the consortium, according to press specultion, but a formal approach is yet to be made.
In March, the supermarket group reported its first annual loss in its history. The£312.9 million plunge into the red was blamed on the cost of converting the final 160 Safeway stores to the Morrisons fascia.
However, TNS research released last week revealed sales up 2 per cent at Morrisons - the first increase since its acquisition of Safeway. Its market share remained unchanged at 11.3 per cent, which was still down on last year's 11.7 per cent.
Last month, ex-Heineken chief executive Marc Bolland was appointed chief executive of Morrisons, ending months of speculation about who would succeed the incumbent Bob Stott. Bolland has left Heineken and will take up his role at Morrisons on September 1.