The newspaper said CD&R was understood to have been preparing equity and debt financing for a counter-bid which could come in the next few days.
If successful, CD&R would open Morrisons convenience stores at fuel stations operated by Motor Fuel Group, which the firm owns, and it would work alongside the existing Morrisons management team, the Sunday Times said.
A spokesman for CD&R declined to comment on the report.
Earlier this week, British money manager M&G joined criticism of the Fortress-led bid that Morrisons has agreed to, which is worth about 6.3 billion pounds and topped a rival offer from Apollo.
Morrisons’ largest shareholder Silchester has also said it is not inclined to support the offer for Britain’s fourth-largest supermarket chain. Schroders (LON:SDR) Plc is also an investor in Morrisons.
SoftBank Group Corp controls Fortress