The scale of the potential impact of David Cameron's success or failure in negotiating key European Union reforms has been underlined by a poll.
As the PM prepares to push his case at a Brussels summit tomorrow, the poll found a comfortable majority - by 56% to 35% - in favour of staying under the existing rules, which the Prime Minister says are unacceptably skewed against UK interests.
An entirely successful renegotiation would harden that support to 65% against 26% - with even 20% of Ukip voters saying they would support continued membership.
But the survey, by ComRes for the pro-reform Open Europe group - suggested that if the PM was unable to persuade other member states to adopt some of the changes he is demanding, opinion would swing dramatically towards a so-called "Brexit".
If there is no agreement on action to prevent non-eurozone countries being disadvantaged by decisions taken by the 18 which use the single currency - one of the four central demands being pushed by the PM - the "Leave" camp would move narrowly ahead, by 46% to 45%.
And failure to restrict new EU migrants' access to in-work benefits for four years - by some distance the hardest of the four to secure - would produce a nine-point swing away from the "Remain" camp, cutting its advantage to just three, 48% to 45%.
The president of the European Council Donald Tusk cautioned yesterday that EU leaders are still "far from an agreement" on the UK's renegotiation demands and insisted there can be "no taboos" when the issue is debated.
The warning, in a letter to the 28 leaders setting the scene for the two-day gathering, comes amid growing pressure on David Cameron to secure concessions.
A cross-party committee of MPs has questioned whether any reform deal could be binding without treaty change, while polls have suggested the outcome of the renegotiation could sway the result of the looming in-out referendum.
Mr Cameron's agenda risks being overshadowed at the summit by controversy over proposals for dealing with the migrant crisis - including new powers for the European Commission to send in specialist squads to member states judged to be failing to control external borders.
The UK's renegotiation demands are due to be discussed by leaders over dinner, with the president highlighting the need to "pave the way for an agreement" at a further council meeting in February.
The key sticking point is believed to be Mr Cameron's call for a four-year ban on migrants claiming in-work benefits in the UK.
The proposal was included in the Conservative Party's general election manifesto, but other countries have flatly dismissed the idea and the Prime Minister has indicated he is willing to consider alternative suggestions.
Number 10 dismissed as "utter nonsense" a report that three cabinet ministers - Home Secretary Theresa May, Work and Pensions Secretary Iain Duncan Smith and Justice Secretary Michael Gove - were among senior figures demanding a tougher stance.
"The Cabinet fully supports the clear manifesto pledge to negotiate a better deal for Britain in the EU and hold an in-out referendum before the end of 2017," a source said.
Open Europe co-director Raoul Ruparel said: "The poll underlines the importance of David Cameron securing a renegotiation package which is seen as credible by swing voters.
"Putting in place measures to protect the UK from being disadvantaged by decisions taken in the eurozone and restricting new EU migrants' access to benefits are particularly crucial, and failure to achieve these objectives hugely increases the chances of a Leave vote.
"The EU should be aware that failing to find a deal on the UK's key demands will increase the chance of a Leave vote in the referendum.
"Similarly, those leading the Remain campaign should be cautious about dismissing the reforms as unimportant, as this poll suggests they matter to the crucial undecided voters."
Protecting non-euro states and strengthening the single market were seen as the most important objectives, with 76% citing those compared with 74% who said curbing benefit claims.
Far fewer were concerned with giving national parliaments a collective veto (69%) or ensuring the UK was not constrained by the EU's stated aim of "ever-closer union" (60%).
Conservative supporters are the most volatile, the poll found.
ComRes interviewed 1,001 British adults via telephone between December 11-13. Data was weighted by age, gender, region and socio-economic grade to be representative of all GB adults.