Morrisons has flatly rejected the proposed offer, saying it "significantly undervalued Morrisons and its future prospects".
City analysts believe CD&R is highly unlikely to make a hostile offer as it will need the highly respected team of Morrisons' chief executive, David Potts, chairman, Andy Higginson, and chief operating officer, Trevor Strain, to stay on board.
All three have been credited with doing an excellent job at feeding the nation during the pandemic.
CD&R is seen as a big, established player in private equity and analysts said it was serious about its offer and has spent millions of pounds on it.
Morrisons owns a huge percentage of its store estate - about 85 per cent is freehold, which is way more than the average of 60 per cent among rivals. This means that a lot of Morrisons' value is tied up in its estate.
Analysts said Morrisons wouldn't want to see its estate sold off as one of the reasons why Morrisons is inherently strong is because it hasn't sold its property.
The Morrisons family own around 5 per cent of the firm so they would not have enough clout to prevent a higher offer getting approval from shareholders. However, many institutions will want to see a 30 to 40 per cent premium on Morrisons' current value.
Shares in Morrisons, which are down 5.5 per cent over the last year, closed on Friday at 182p, valuing the group at £4.33bn. A 30 per cent premium would be a bid of £5.63bn and a 40 per cent premium would be a bid of £6.06bn.
Morrisons declined to comment on what will happen next, but city analysts said it is too early to say whether CD&R will increase its offer or walk away.
A spokesman for Morrisons referred the Yorkshire Post to the company's statement which came out on Saturday evening.
The statement said that Morrisons received the "unsolicited, highly conditional non-binding" proposal of 230p a share on Monday and rejected the proposal on Thursday.
The group said: “The board of Morrisons evaluated the conditional proposal together with its financial adviser, Rothschild & Co, and unanimously concluded that the conditional proposal significantly undervalued Morrisons and its future prospects.
”Accordingly, the board rejected the conditional proposal on 17 June 2021."
CD&R had earlier said it was considering a possible cash offer for Morrisons.
Under takeover rules, CD&R has until July 17 to announce a firm intention to make an offer.
CD&R’s approach underlines private equity’s growing appetite for UK supermarket assets, attracted by their cash generation and freehold assets.
In February, brothers Zuber and Mohsin Issa and private equity firm TDR Capital purchased a majority stake in Asda from Walmart in a deal valuing the UK supermarket group at £6.8bn.
Sky News said a formal bid from CD&R could involve Terry Leahy, the former Tesco CEO, who is a senior adviser to CD&R. While he was at Tesco, Mr Leahy oversaw Mr Higginson and Mr Potts.
Morrisons has about 500 stores and 118,000 staff, making it one of the country’s biggest private sector employers.
In March, the group reported a halving of annual profit due largely to costs incurred during the Covid-19 pandemic, but it forecast a bounce back in its 2021-22 year.