The former head of Britain’s intelligence agency MI6 has warned of a “populist uprising” unless leaders can show that they have control of the migrant crisis in Europe.

Sir Richard Dearlove, who ran the agency from 1999 until 2004, warned that the numbers of migrants arriving in Europe could be in the millions and cautioned against offering visa-free travel to the EU for Turkish citizens.

“If Europe cannot act together to persuade a significant majority of its citizens that it can gain control of its migratory crisis then the EU will find itself at the mercy of a populist uprising, which is already stirring,” he added.

The controversial comments — made to the BBC on Monday — come in a highly-charged political climate just five weeks away from the referendum on membership of the EU. Sir Richard, who was also Master of Pembroke College, Cambridge, from 2004 to 2015, said the Brexit debate is “the first roll of the dice in a bigger geopolitical game”.

The EU is struggling to respond to a surge of desperate migrants that has resulted in more than 3,000 deaths, with many perishing or missing in the Mediterranean. The surge of migrants and refugees, the largest movement of people Europe has seen since 1945, has raised doubt about open borders and provoked a dispute over sharing the burden

One in seven asylum seekers in the EU’s flagship scheme to relocate refugees throughout the bloc has either refused to be moved or “absconded”, according to figures provided by the Greek government.

The policy, agreed last year after considerable acrimony, was designed to spread more evenly the EU’s refugee burden by relocating 160,000 asylum seekers from frontline states Greece and Italy to other members of the bloc.

[Giving Turkish citizens visa-free movement is] perverse, like storing gasoline next to the fire we’re trying to extinguish

Meanwhile, Turkey and the EU are arguing over a package agreed by EU leaders in exchange for Turkey’s help as they scramble to prevent a repeat of the chaotic scenes of last year, when more than 800,000 refugees and migrants paid smugglers to put them on boats to Greece.

Sir Richard said giving Turkish citizens visa-free movement which would be part of the deal was “perverse, like storing gasoline next to the fire we’re trying to extinguish”.

Earlier this month, Britain won a carve-out from new EU rules on asylum seekers that will allow the UK to continue expelling migrants that arrive via other EU countries.

Under plans announced by the European Commission to overhaul Europe’s much-criticised rules on asylum seekers, most EU countries would be required to participate in a new quota system that would force them to accept migrants if a front-line state becomes overwhelmed, as Italy and Greece were at the height of the refugee crisis. Failure to comply would result in fines of €250,000 per person.