Jeremy Hunt has raised the prospect of a new EU referendum while announcing himself as a possible candidate to be the next leader of the Conservative Party.

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.

Cameron on Brexit: Will of the people to be respected

“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.

Brexit reactions – in pictures Show all 10 1 /10 Brexit reactions – in pictures Brexit reactions – in pictures Supporters of the Stronger In campaign look at their phones after hearing results in the EU referendum at London's Royal Festival Hall AP Brexit reactions – in pictures Leave supporters cheer results at a Leave.eu party after polling stations closed in the Referendum on the European Union in London Reuters Brexit reactions – in pictures Mr Cameron announces his resignation to supporters Getty Brexit reactions – in pictures Donald Tusk proposes that the 27 remaining EU member states ‘start a wider reflection on the future of our union’ Getty Brexit reactions – in pictures Ukip leader Nigel Farage greets his supporters on College Green in Westminster, after Britain voted to leave the European Union PA Brexit reactions – in pictures Supporters of the Stronger In Campaign react as referendum results are announced today Getty Brexit reactions – in pictures Boris Johnson leaves his home today to discover a crowd of waiting journalists and police officers Getty Brexit reactions – in pictures Leave EU supporters celebrate as they watch the British EU Referendum results being televised at Millbank Tower in London Rex Brexit reactions – in pictures Supporters of the Stronger In Campaign react as results of the EU referendum are announced at the Royal Festival Hall Reuters Brexit reactions – in pictures Supporters of the Stronger In campaign react after hearing results in the EU referendum at London's Royal Festival Hall PA

“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."

Mr Cameron’s final appearance at a Brussels summit was due on Tuesday (GETTY)

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.