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 child suffered months of headaches that left doctors baffled. He was finally diagnosed after he drew an Etch-A-Sketch of his condition.
A guarantee from City Hall to bankroll the annual £2-3m bill was a key planning condition imposed by Westminster and Lambeth councils.