The shadow chancellor claimed that, according to the UK constitution, Labour must be offered the chance to govern if Ms May is no longer able to command a majority in the Commons.
Until now, Labour’s policy has been that a general election should be triggered if the government’s proposed withdrawal agreement is rejected by parliament, but Mr McDonnell admitted this would be “difficult” to bring about.
Instead, he said, Mr Corbyn should be given the chance to put Labour’s plans before MPs, adding that he believed they would be backed by the Commons.
Ms May has faced questions over the status of her governing majority, with the Democratic Unionist Party, whose votes she relies on, refusing to support the prime minister unless she changes her Brexit plan.
Speaking to City leaders at an event organised by Reuters, Mr McDonnell said: “We haven’t sufficiently explored the concept of a minority government in our constitution, our custom and practice.
“If [the government] can’t command a majority, usually it is then the duty of the monarch to offer to the opposition the chance to form a government – and that would be a minority government – to see whether they have a majority in parliament.
“I think we could secure a majority in parliament for some of the proposals that we’re putting forward.”
Mr McDonnell was speaking after the DUP temporarily pulled its support for the government in protest at Ms May’s Brexit deal.
The Northern Irish party abstained on a number of key votes on the Finance Bill, throwing into doubt the future of its Commons pact with the Conservatives.
The prime minister will need the DUP’s votes if she is to have any chance of getting her proposed deal through parliament.
Mr McDonnell said the real test would come when MPs are given a “meaningful vote” on the Brexit deal next month.
He said: “The process around offering the opposition party [the chance to try to form a government] usually comes as a result of the minority government losing votes within the House.
“That’s already begun to happen on the Finance Bill, and the Finance Bill is usually the most significant piece of legislation in any year.
“However, I think the test is whether or not the government is losing consistent votes on the issue of the deal itself. If it is, then it’s demonstrating that as a minority government it hasn’t got the will of the House, the majority of the House, and therefore it’s a situation where it’s ungovernable, they’re not in government.
“Therefore the normal process is that the opposition party should be offered the opportunity to see if it can form a government that could secure a majority in the House for its proposals, and I think that’s the process we should go through”.
Turning to Brexit, the shadow chancellor said he was confident that parliament would block a no-deal outcome “at all costs”.
He said: “There is an overwhelming majority opposing anything that smacks of being no deal.
“I think that the concerns that people have of falling off a cliff edge – there is no majority for it. There will be a majority against it.”
Mr McDonnell refused to confirm whether he thought Brexit would happen at all, saying that while Labour wants to respect the result of the referendum, “it’s very difficult to predict at the moment what the final outcome will be”.
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