THOUSANDS of migrants in Calais could move within hours to Britain if the public votes No in the forthcoming referendum on EU membership, No 10 has suggested.

With June 23 regarded as David Cameron’s target date for a poll, the indication from Downing Street is that a “huge number” of those encamped in northern France in the so-called “Jungle” camp could begin to move “effectively overnight” and set up camps in Folkestone and at other entry points in southern England.

“The point that is being made here is that we currently have these juxtaposed controls with France that, should the UK leave the EU, there’s no guarantee those controls would remain in place and if those controls weren’t in place, then there would be nothing to stop thousands of people crossing the Channel overnight and arriving in Kent and claiming asylum.”

Asked how quickly migrants could start crossing the Channel, the Prime Minister’s spokesman said: “It could happen very quickly.” Asked if it could be within hours or days, he shrugged his shoulders and said: “I can’t put a date on it.”

Eurosceptics have reacted angrily to the claim that originated from Whitehall, saying it was “blatant scaremongering”. But No 10’s reaction is set to increase the controversy.

Nicola Sturgeon, who will be campaigning for Britain to stay in the EU, has warned Mr Cameron from instigating another “Project Fear”, which the UK Government, according to its opponents, placed at the forefront of the No campaign in the Scottish independence referendum.

The First Minister has warned against repeating the exercise as in the independence referendum the Yes camp made up 25 points in the polls. With regards to the EU referendum, the polls are much tighter; indeed, one last week had the Leave camp several points ahead. Ms Sturgeon's fear is a campaign of scaremongering from Whitehall could lose the vote to the Brexit campaign.

Tory MP Sarah Wollaston said the PM’s claim was "simply not credible" and complained pro-EU campaigners were taking voters "for fools".

"Ratcheting up the alarmist rhetoric on security by Project Fear In campaign will backfire; people don't like to be be taken for fools," she posted on Twitter.

"It is simply not credible to claim that EU co-operation on security issues would end in the event of Brexit,” she added.

The PM’s spokesman pointed to the treaty with France to have UK border controls in France but that there was “no guarantee” that that would continue if Britain voted to leave the EU, which would throw the whole security relationship into question.

Asked about scaremongering, he said this was about “raising a genuine concern” about high levels of migration and that the focus was still on the renegotiation and getting the best deal for Britain.

In a key development in the referendum campaign, the PM is now preparing to make national security issues a key plank of his campaign to keep Britain in the EU over the coming weeks.

He will say that so-called "Brexit" would see France pull out of the 2003 Le Touquet treaty, which means that checks for migrants trying to stow away on lorries or trains heading for Britain are carried out in Calais.

"The French would love to pull out of the arrangement," a senior source told the Daily Telepgraph. "We will be telling people - look, if we leave the EU the Jungle camp in Calais will move to Folkestone. That is not something people want."

The claim was dismissed as "scaremongering" by Vote Leave campaign chief executive Matthew Elliott.

"This is blatant scaremongering from Number 10 that has no grounding in reality," he said.

"UK border controls are in France because of a bilateral treaty, not because of our EU membership, and a result of the camps in Calais, not the cause of them. Clearly, No 10 is in a blind panic over the failing renegotiation."

The latest spat erupted amid continuing speculation as to which way some senior Conservative Party figures would vote once the referendum campaign is under way.

Justice Secretary Michael Gove - who is said to be torn between loyalty to Mr Cameron and an instinctive desire to leave - appeared to be keeping a low profile, avoiding journalists' questions, despite a major Government launch on prisons policy.

London mayor Boris Johnson - who has been courted by both camps - was also keeping observers guessing, insisting he would like to remain in a reformed bloc but would "wait until you see the whites of their eyes" before making his decision.

"This is the moment to stiffen the sinews, summon up the blood, squint down the barrel and only when you see the whites of their eyes should you finally let fly and decide whether to stay or leave the EU; because the arguments are as finely balanced as they have ever been," he wrote in his column in the Telegraph.

Meantime, Rob Whiteman, a former chief executive of the UK Border Agency, said Mr Cameron was entitled to argue that France would "almost certainly" end the Le Touquet treaty if Britain left the EU, resulting in a big increase in the number of migrants entering the UK.

"There has been lots of upsides for the UK since the treaty was negotiated in 2003, not much upside for the French," he told the BBC Radio 4 Today programme.

"If you're found in a concealed vehicle, you're a clandestine, once you set foot on British soil you can claim asylum whereas if you're found in a vehicle on French soil you can't claim asylum in Britain.

"Before that treaty was put in place asylum claims were running at 80,000 a year in the UK. They are now running at about 30,000 a year so we would probably see, let's say, another 50,000 asylum claims a year which we used to get before the treaty came in."

Meanwhile Margaret Thatcher's authorised biographer, Charles Moore, said that after she left office, the former prime minister had argued that Britain should leave the EU.

Over the weekend one of her closest aides - her foreign policy adviser Lord Powell of Bayswater - said he believed that she would have backed Mr Cameron's re-negotiation plan.

Mr Moore declined to speculate on how she would have voted in the referendum if she was still alive, but said that she had told him that she thought Britain should withdraw.

"While the whole time she was in office she supported our membership, though with increasing lack of enthusiasm, after she left office she privately thought we ought to leave and told me that and told that to quite a lot of people," he told the Today programme.

Elsewhere, Liberal Democrat leader Tim Farron said: "The leave campaigns simply refuse to acknowledge the threat that turning our back on Europe would have to our national prosperity and security. They have no plan, and no idea of the turmoil it would cause.

"But we must not trade blows based on fear, or wilfully stoke a hazardous row on immigration.

"Rather than turning people who are fleeing war and persecution into a political pinata we should all focus on what can be done to help.

"We should be confident in our ability to make the case that by remaining in, every family, every business, and every person in Britain is part of a stronger, safer and more prosperous nation."