Boris Johnson says David Cameron should remain Prime Minister if the UK votes for Brexit

While campaigning for Vote Leave the MP also repeated the disputed claim that the UK sends £350 million a week to Brussels, and refused to apologise for his comments about Barack Obama

The PM's friendship and rivalry with Mr Johnson stretches back to their days at Eton and Oxford (Getty Images)

Boris Johnson has denied a Leave vote in the European referendum would clear the way for him to become Prime Minister.

Many in Westminster believe Mr Johnson hopes a vote to pull out of the European Union in the June poll would force Mr Cameron to step down – and is, in part, why he is supporting Brexit.

But asked directly whether he thought Mr Cameron should remain in 10 Downing Street to oversee negotiations for Britain's withdrawal from the EU, the former London mayor told ITV1's Good Morning Britain: “Yes, absolutely. Of course he can, and I think he must.”

Mr Johnson was conducting a series of interviews to kick off a Vote Leave battlebus campaign in Truro, Cornwall.

In them he refused to apologise for comments he made about President Obama’s attitude to Britain being based on his “part-Kenyan” heritage and “ancestral dislike of the British empire”.

And controversially he repeated the Vote Leave claim that the UK sends £350 million a week to Brussels - despite a second warning from the head of the UK Statistics Authority Sir Andrew Dilnot to desist.

Sir Andrew wrote to the campaign on Tuesday to tell them it was “disappointing” that they continue to use a figure which he believes is “potentially misleading”.

Mr Johnson - widely regarded as a likely contender for a Tory leadership battle when Mr Cameron stands down - insisted that the Brexit camp could “guarantee” that Britain's EU contributions would be spent on UK priorities like Cornwall's fishing and farming industries if the UK votes to leave.

He told BBC1's Breakfast: “Of the £20 billion we send to Brussels a year, £10 billion we never see again. It goes on all sorts of things - Greek tobacco farming, Spanish bull-fighting.

”With that net money back in our country we could fund things like the NHS, our science base, our academic health science centres even more generously than we currently do. That argument just doesn't stack up.“

Mr Johnson was not chastened by US president Barack Obama's warning that Britain would be ”at the back of the queue“ for a trade deal if it left the EU.

He told Good Morning Britain: ”Obviously, when the US wants us to be at the front of the queue for various things - the Iraq War - then that's a different matter.

“Most sensible people will recognise that we will do a free trade deal not just with the EU, but we will have the opportunity for the first time in 43 years to do free trade deals not just with America but with India, China, Australia and New Zealand, which we currently cannot do because we are a member of the European Union.”

He rejected the claim of Labour's Alan Johnson that Brexit campaigners were “extremists” who could see nothing good in Europe, telling the BBC: “I do think it's very odd that we are being called extremists and irrational when only the other day we were told World War Three was going to break out if we voted to Leave. That cannot be sensible.

”Everybody knows that peace in this continent is really guaranteed by Nato. If it really is true that World War Three and bubonic plague are about to break out, why on earth are we having this referendum?

“I love Europe. I have many happy memories of living, working, going on holiday to Europe. Most of my family come from one European country or another.

”But there's a difference between Europe and the European institutions, and they are now evolving in a way which is not compatible with the long-term health of our democracy.“