At a meeting with M&S chief executive Stuart Rose and chairman Paul Myners, Green and a Goldman Sachs representative proposed an increased cash bid of 370p per share. The offer was worth £8.3bn in total.
Green’s earlier cash bid of between 290p -310p per share - worth £7bn - was slapped down earlier this month.
In a statement issued to the stock market this morning, M&S said it considered that the offer "significantly undervalued the group and its prospects".
M&S shareholders are believed to have wanted a bid of at least 400p per share.
Myners reportedly later told the press, "This business is not for sale."