The Health Secretary said he was “seriously considering” running against Boris Johnson on ITV’s Good Morning Britain programme as the fall-out from the Brexit and David Cameron’s resignation continued.
Amid calls by some Remain voters for a second EU referendum, he said another vote would be one of several ways to ensure British people backed the terms under which Britain will leave the union.
“I don’t think there (necessarily) needs to be another referendum but I think there needs to be some kind of democratic endorsement,” Mr Hunt, who backed Remain, said.
“We’ve had nine general elections since we joined the EU and this is such a big thing that I think the terms under which we leave the EU also need to be put to the British people.”
When asked whether he would fulfil the Leave campaign’s pledge to spend £350 million of weekly EU contributions on the NHS if he became Prime Minister, Mr Hunt was cautious.
“I was sceptical about that number and the reality is what matters far more than contributions to the EU is the state of our economy,” he said.
“The reality is we are going to have very choppy times economically and I don’t think we should be divvying that money up.
“I am someone who thinks that the NHS is our most important public service, it’s the one that the public values the most.”
Mr Hunt, who has previously faced intense criticism for his handling of the junior doctors’ contract dispute, said he had “until Thursday lunchtime to decide” on whether he will join the Conservative Party leadership race.
In April, he said his post as Health Secretary was likely to be his "last big job in politics" as junior doctors started one of several rounds of strikes.
Writing in the Daily Telegraph, he previously said the public did not vote on the conditions of the country's departure from the EU and has called for a deal to be negotiated, then put before the electorate.
"We must not invoke Article 50 straight away because that puts a time limit of two years on negotiations after which we could be thrown out with no deal at all,” Mr Hunt wrote.
"So before setting the clock ticking, we need to negotiate a deal and put it to the British people, either in a referendum or through the Conservative manifesto at a fresh general election."
He is in favour of a Norway-style model of access to the single market - but with the addition of “sensible restrictions” to freedom of movement.
He also called for all the prospective Tory leadership contenders to promise that the current 110,000 EU citizens working in the UK would be granted permanent residence.
Mr Cameron, who will remain Prime Minister until the Tory party conference in October, has ruled out a second referendum on Britain’s membership of the European Union.
Almost 4 million people have signed a petition calling for a change to referendum rules forcing a new vote but a Downing Street spokesperson said the prospect was “not remotely on the cards”
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