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".
More top news
How is Sainsbury's positioned to tackle the challenges of low-cost rivals and changing shopping habits?
The head of football's anti-discrimination body Kick It Out has responded to attacks by QPR defender Rio Ferdinand in his autobiography
Prime Minister David Cameron and his wife Samantha surprised diners in Solihull by dropping into fast food chain Five Guys for burgers.