Denmark is preparing to hold a referendum on its relationship with Europe on Thursday - a vote that could hold crucial lessons for David Cameron when it comes to selling his own renegotiation with Brussels to ordinary British voters.

The Danish vote on whether Denmark should give up a host of EU opt-outs on justice and security has provided a lightning rod for public questions about border security, uncontrolled migration and the ability of the European Union to govern itself.

“More EU? No thanks!” is the campaign slogan from the Danish People’s Party (DPP), leading a no campaign which is being watched with trepidation in Downing Street.

It is a slogan that will resonate with many Danish voters, whose shock rejection of Maastricht in 1992 was the first big blow to European integration.

Polls currently show the two sides neck and neck, but with Europe facing so many existential crises over the euro, migration and border security, the vote is already being cast as a confidence motion in the EU and the Danish political elite.

In what could be a harbinger of how the referendum battle will play out in Britain next year, the nitty-gritty of policy has largely been ignored in favour of broad-brush scaremongering - on both sides.

For the yes camp, campaigners have pulled no punches, plastering pictures of victims of child sex abuse and human trafficking across the media to convince voters that if they don't vote in favour, Denmark will be pushed out of organisations such as Europol, thereby endangering its security.

“Strengthen Denmark’s Police. Vote yes” said one yes campaign poster, along with a retro picture of a blonde policeman and Hamlet’s castle in Elsinore flying the beloved Dannebrog flag.

Peter Skaarup, deputy leader of the Eurosceptic Danish People’s Party, accused the yes side, led by Lokke Rasmussen, Denmark's prime minister, of running a “scare campaign”.

“It’s rubbish. Who doesn’t want to fight paedophiles? Who doesn’t want to fight burglary gangs? But that’s not what this referendum is about,” he said to The Telegraph at a rally with party loyalists in Copenhagen last week.

What Danes are actually voting on is whether to convert Denmark’s "opt-out" exemption from EU justice and home affairs legislation into a pick-and-choose "opt-in" which - like that of the UK and Ireland - will allow its parliament to enact the legislation it wants and reject much of the rest.

Denmark’s "Yes" parties have picked out 22 pieces of EU legislation they definitely want to enact, ranging from police cooperation to divorce law, and cross-border debt recovery, and 18 pieces of EU legislation, including unpopular areas of immigration and asylum law, they commit not to.

The opt-outs on justice and security are a sensitive issue in Denmark where popular opinion believes there has been too much encroachment from Brussels on issues – like security – that were previously considered areas of sovereign control.

Six out of nine Danish political parties, and a good two-thirds of MPs, now support ending the opt-out, but the voters – who are generally more Eurosceptic than their politicians - remain split roughly 50:50.

Overall, the two sides remained neck and neck, with a weighted average of all polls this month published on Monday giving the yes side a slender lead, with 51.1 per cent of voters saying yes and 48.9 per cent saying no.

A separate poll found that 20 per cent of those surveyed were “more likely” or “much more likely” to vote yes after the Paris attacks.

However, a poll by the Danish polling agency Megafon on Monday showed the no side seeming to gain a late lead, with 40 per cent of respondents saying they would vote no, 35 per cent saying they would vote yes, and 25 per cent undecided.

Analysts are divided on what a no vote would mean for the British referendum, with some reading it as a warning of the risk of calling referendums in Europe in which publics reject the advice of political elites in Brussels and their own national parliaments.

There are fears among UK officials that a no vote would also be pounced on by British Eurosceptics as a rejection of Mr Cameron’s own vision for a more flexible “pick-and-mix” Europe where member states would opt in, rather than being co-opted into every closer union.

Others, like Pieter Cleppe of the Open Europe think-tank in Brussels believe a Danish no could be just the warning that Mr Cameron needs as he enters the critical phase of his negotiations with the other 27 member states.

“A ‘no’ vote could help Cameron – it will be a shot across the bows for Merkel and the European Commission and a clear signal that other countries, not just Britain, are demanding a more flexible union,” he said.

In the end, the outcome could come down to trust: a poll at the start of the campaign showed that only 14 per cent of Danes believed that voting no would actually see Denmark excluded from the EU police agency as the Danish prime minister and the yes campaign have warned.

In a potential boost to the yes campaign, the EU warned Danes that if they vote no, they will not necessarily be offered parallel agreements allowing them to participate in Europol and other justice areas – a move seen as exemplary of EU bullying and inflexibility in the no camp.

“There is no trust in the government in EU affairs,” said Jens-Peter Bonde, a veteran of Danish no campaigns and former eurosceptic MEP. “There’s a long list of promises in referendums which have been broken afterwards, and people remember when they’re not told the truth.”

The Danish People’s Party has sought to exploit such mistrust. “If you vote ‘yes’, the risk is that we will not be able to continue as a sovereign country,” Mr Skaarup warned darkly after his rally on Thursday.

“If you vote ‘yes’, you step into a possibility of bringing asylum policy into the EU decision-making system — and if you do that, no guarantee whatsoever could prevent us entering a common immigration policy.”

Ian Manners, a politics professor at Copenhagen University and expert on the opt-outs, also believes the Danish People’s Party will win the argument.

“I can’t see anything other than a ‘no’,” he said, “because the ordinary people, even the ordinary politicians, have no clue about what this is about.”