The UK takeover panel said the firm, which had an original £5.5 billion approach rebuffed, has until 5pm on August 9 to make a firm offer.
Since turning down CD&R’s proposals, Morrisons’ management have agreed a £6.3 billion takeover by a consortium involving private equity rivals Fortress and Apollo.
It is understood that CD&R is working on a financial package to allow it to put forward a second bid for the retailer.
The new deadline for an offer is seven days before Morrisons shareholders will vote at a general meeting on the current Fortress-backed deal.
Apollo Management had initially been in the frame to compete in a three-way bidding war but decided to join forces with Fortress earlier this week.
Shares in Morrisons are currently at 265p, reflecting a feeling among investors that it is likely to be sold for more than the 254p per share deal agreed with Fortress.
On Thursday, Morrisons also published bid documents which show that bankers and advisers are in line to take home around £275 million if the proposed takeover goes ahead.
Advisers could receive around £169 million for financing the deal, with bankers set to gain £36 million for broking advice for Fortress, as well as £17 million in legal advice.
Morrisons would also pay £42 million to advisers involved its process.