The Chancellor of the Exchequer, in Cannes for the European summit, predicted that Mr Redwood would stand on an ideological right-wing platform which would see Britain "adopt attitudes towards Europe which would detach us from the European Union" - and lose the country huge benefits.
"If he abandons the healthy scepticism of John Major, the opt-outs towards a single currency, I don't think the Conservative Party is leadable on that basis," Mr Clarke said in an interview on BBC Radio's Today programme.
Michael Heseltine, still heavily tipped to enter the ring in a second ballot, said Mr Major deserved a good deal better than a leadership challenge from the former Secretary of State for Wales.
Speaking on a factory visit in Eastleigh, Hampshire, the President of the Board of Trade said: "We are all of us part of his team, we have all worked on the same policies, taken the same decisions and I think he was entitled to expect that we would stand with him.
"Certainly everyone else in the Cabinet has made that very clear, some of us over a considerable period of time."
Douglas Hurd, the outgoing Foreign Secretary, said from Cannes that the party faced a stark choice.
"Are we going to go to the country on the basis of some right-wing and extremist agenda or are we going to go to the country on the solid centre policies which the Prime Minister has worked out?
"We can go off into a right-wing, extreme agenda, forget the solid centre, do really what Michael Foot did to the Labour Party - rally it round a set of policies after 1979 which warmed the hearts of a good many Labour people and made absolutely sure that they lost the next election."
The Prime Minister needed a sufficiently strong vote to restore his authority. "He will need to be able to govern with authority," Mr Hurd said.
Emerging from a lunch with industrial correspondents, Michael Portillo, Secretary of State for Employment, said: "I am very sorry to lose him [Mr Redwood] as a Cabinet colleague." He added: "The Prime Minister is my candidate and that remains the case."
Mr Portillo refused to be drawn on whether he would enter the contest in a second round, but was being strongly urged by his supporters to do so.Reuse content