David Cameron is not scaremongering by warning that the French border and migrant camps could move from Calais to Dover in the event of the UK leaving the EU, according to Sir Peter Ricketts, who was British ambassador in Paris until last month.

Ricketts, who also served as Cameron’s national security adviser, said he judged it likely the French would review its position – on the Anglo-French 2003 Le Touquet treaty, which allows for UK border guards to be stationed in northern France, in the event of Brexit.

Downing Street’s claims that the Jungle refugee camp in Calais could move to Kent have been ridiculed by Eurosceptics. They point to comments by Bernard Cazeneuve, the French interior minister, who several months ago said France did not want to tear up the treaty governing the border.

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Ricketts told BBC Radio 4’s Today programme: “This is a bilateral treaty but it was made in a multilateral context. If the context changed and Britain made a major decision to leave, I think it very likely France would review its position as well.

“Bear in mind what France is doing. It’s got 1,000 of its crack riot police deployed in Calais – far more than in Marseille. They are bottling up 7,000 or more migrants in the camps around Calais. They are taking a lot of political flak for that, there is a lot of humanitarian pressure on them.

“They are doing it effectively to protect our border. We get a secure border and the French carry a lot of the load. They are doing it because they see us as a very important ally in the EU. If that stopped, the incentives change for France.”

Ricketts said there were already calls from the French opposition to renege on the treaty.

On Monday Cameron said the UK must give France “no excuse” to tear up the treaty. The prime minister’s comments infuriated Eurosceptics, who warned that pro-Europeans would seek to imitate the so-called “project fear” tactics of the Scottish referendum. Liam Fox, the former defence secretary, said he was “sad and disappointed” by Cameron’s remarks.

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A No 10 spokesman said: “We currently have these juxtaposed controls with France that, should the UK leave the EU, there is no guarantee that those controls would remain in place. 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.”

He added: “We have an arrangement in place with France. We are both EU partners. Should we leave the EU there is no guarantee that the relationship, in terms of the controls we have in France at the moment, would continue. If those controls didn’t continue then there are thousands of people there who are there specifically because they want to come to the UK who would then come to the UK.”

Pro-Europe Tories will intensify the pressure on the anti-EU camp with a warning that a UK exit would mark a “leap in the dark”.

Nick Herbert, the former justice minister who chairs the campaign group Conservatives for Reform in Europe, will say: “The personality differences on the leave side are clear for all to see. But they also have serious policy disputes. Despite years of campaigning for Brexit, they can’t even agree among themselves what the alternative to EU membership should be.”

The pro-EU camp believe it is on firm ground because the anti-EU camp cannot guarantee what deal the UK would sign with the EU after a vote to leave. Under the terms of the Lisbon treaty, a country leaving the EU loses its seat on the European council and has to negotiate within two years with two member states appointed by the remaining 27 EU leaders.