Carr, who announced his departure from Cadbury in February, is said to be looking abroad for his next corporate post.
His withdrawal creates more uncertainty for the high street giant, which has installed a new chief executive, Marc Bolland, and lost its chief financial officer Ian Dyson in the space of three weeks.
The company is expected to announce profits upwards of £620m tomorrow, a year-on-year increase of four per cent.
The retailer is said to be keen to announce a new chairman before its annual meeting on 14 July. Rose continues to run the company day to day while Bolland completes his induction, but is unlikely to remain past the end of the year.
Still thought to be in the frame for the top job is Alan Parker, outgoing chief executive of Whitbread and lecturer on international hotel management at the University of Surrey.
Also said to be a contender is Lord Mervyn Davies, who left Standard Chartered last year to become a life peer and trade minister in Gordon Brown’s government.
Others in the running include Sir Crispin Davis, former chief executive at Reed Elsevier, and deputy chairman of Thomson Reuters, Niall Fitzgerald. Sir Victor Blank, former chairman at Lloyds, has also been mentioned.
An M&S spokesperson said: “The chairman recruitment process is proceeding well.”