Former NHS Trust chairman Roy Lilley said it has been suggested to him that Sir David Nicholson will "be gone before September".
He told Daybreak: "He will be gone before the party conferences, they won't want to take this to the conferences".
On the subject of Sir David's expenses he said, "you would expect a guy like that to have high expenses", he added, "but there are some 'unexplained' expenses".
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The heavy showers will ease off this weekend.
Labour MPs are attempting to build a consensus for a single challenger to Corbyn, while the big game in the Tory Party is Stop Boris.
Hefty showers and longer, slow moving downpours into this evening and again on Saturday - some lively with thunder.