The news will come as a blow to Rose, who is understood to have been in talks with Green, a friend and former colleague, for a number of months.
The two men are believed to have met to discuss the job after Christmas, and Green is understood to have had two additional meetings with M&S senior management more recently.
Last November, Green set out ambitious plans for Tesco's fashion range to rival high street giants M&S, Next and Topshop. He said that Tesco would invest millions in an expanded range of upmarket clothing to be sold in designated areas of its supermarkets.
Tesco boss Sir Terry Leahy is understood to have offered Green a more powerful role with the company, along with a large pay rise, to prevent his departure.
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