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David Cameron's earliest date for the EU referendum has been delayed to June 23 to avoid “voter fatigue” after the London mayoral elections.

Downing Street is worried a low turnout will favour the campaign to leave the European Union because “antis” are thought to be more highly motivated.

A big turnout of Londoners in particular is seen as crucial to Mr Cameron winning a vote to stay — because polls show people in the cosmopolitan capital are more favourable to the EU than other English regions.

Scotland’s first minister Nicola Sturgeon has warned against a referendum in early June for the same reasons. Scotland holds elections to its Holyrood Parliament in May.

A senior Tory told the Standard: “A high turnout is good for the In campaign, which means a date towards the end of June is possible, but nothing earlier.”

A Downing Street source said a decision on a date would only be taken when a deal was struck for EU reform. Talks between Mr Cameron and other EU leaders are at a crucial stage ahead of a summit in three weeks.

Germany’s Angela Merkel has offered Britain the right to halt in-work benefits like tax credits for any EU migrant on less than 20 hours, while continuing to pay the benefits to British part-time workers like single mothers. However, the proposal is seen as unlikely to stem mass migration, and falls short of the Tory manifesto pledge to stop such benefits for four years to all EU workers.

A Whitehall source said the German proposal was “patently discriminatory [against EU workers coming to Britain] and requires treaty change, but is seemingly acceptable because ... well, Mrs Merkel and Germany back it!”

Eurosceptic ministers are cool about the Merkel plan because it falls short of the Tory manifesto pledge for a four-year curb on benefits.

Barack Obama’s polling guru Jim Messina, who advised the Conservatives’ general election victory, is advising the campaign to stay in the EU.

He joins other senior figures from the Tory election machine, including campaign organiser Stephen Gilbert, digital campaigners Craig Elder and Tom Edmonds, field ops expert Stuart Hands and pollster Andrew Cooper.

Campaigners for the referendum clashed today. Businessman Jon Moynihan, for Vote Leave, claimed Britain would get a free trade deal if it left the EU and played down talk of a crisis, saying: “The day we vote to leave the EU, nothing will happen, we will have exactly the same relationship. Then they will enter into negotiation.”