It was picked out of an ashtray by a nurse while the former prime minister was recuperating from a fractured hip at Middlesex Hospital in 1962.
She saved it for her little brother in a bag marked "soiled dressings disposal" where it has been kept since.
Timothy Medhurst of Duke's in Dorchester said it could sell "for thousands" due to "great interest".
Churchill, then 87, was admitted to hospital after falling out of bed while on holiday in Monte Carlo.
He was flown to the UK on an RAF VC10 at the request of Prime Minister Harold Macmillan, and spent weeks in the private Woolavington Wing.
Churchill also contracted bronchitis and pneumonia, which led to thrombosis, but he kept up his smoking habit and eventually recovered.
Mr Medhurst said the cigar marked a "pivotal moment" in Churchill's life.
"Churchill had already experienced many injuries, and to have such a devastating injury occur at such an old age was a threat to his stalwart 'British bulldog' image.
"The operation to fix the fracture would have a caused a shortening in his leg and photographs after 1962 always show Churchill leaning on his iconic walking stick.
"It's not surprising that Churchill's cigar smoking in his hospital bed was widely recorded...the cigar is something we all associate with Churchill."
Churchill once said smoking cigars was like falling in love.
He said: "First, you are attracted by its shape; you stay for its flavour, and you must always remember never, never to let the flame go out."
He died in 1965, aged 90.
His cigar goes on sale on 15 October.
The chewed end of a cigar smoked by Sir Winston Churchill in a hospital bed is expected to fetch £1,000 at auction.