"Hard work" will be needed to reach a deal on Britain's renegotiation of the terms of its EU membership which could be acceptable to leaders of other member states by the target date of February, European Council president Donald Tusk has said.
Prime Minister David Cameron pleaded tonight for his 27 EU counterparts to find a way to address British concerns about immigration after they roundly dismissed a key plank of his renegotiation drive.
Over dinner at a crucial summit in Brussels, described by Mr Tusk as "a make-or-break moment", the Prime Minister suggested the "unprecedented" numbers coming to the UK could result in the public voting to leave the union.
The personal appeal to his 27 counterparts came after they queued up to reject the idea of a four-year ban on migrants claiming in-work benefits.
German chancellor Angela Merkel has said she will not agree to anything "discriminatory" or that runs counter to the principle of freedom of movement, while Mr Tusk indicated the proposal was regarded as "unacceptable".
Mr Cameron has already made clear he is willing to consider alternatives that would reduce migration "pull factors".
But although officials are thought to be working on options - such as an "emergency brake" on excessive inflows or a residency test - no other plans have been formally put on the table.
At a press conference following the talks, Mr Tusk said he would aim to table a text on proposed reforms in the run-up to the next meeting of the European Council in February.
Mr Tusk characterised the discussions over dinner as "substantive and constructive".
"Prime Minister Cameron set out in detail his position, in particular regarding benefits and free movement," said the council president. "He explained his request for a model based on four years and reiterated his openness to alternative solutions only if they could achieve the same objective.
"Leaders voiced their concerns but also demonstrated willingness to look for compromises.
"Building on this positive debate, we agreed to work together to find solutions in all four baskets raised by Prime Minister Cameron. Hard work on all baskets is still ahead of us.
"First we will work closely with the UK and the (European) Commission and in the run-up to the February European Council, I will table a concrete text to all the leaders."
In his 40-minute presentation - described by UK sources as his longest ever speech to leaders at an EU summit - Mr Cameron sought to persuade them that British worries must be accommodated ahead of the looming in-out referendum.
"The levels of migration we have seen in a relatively short period of time are unprecedented, including the pressures this places on communities and public services," he said.
"This is a major concern of the British people that is undermining support for the European Union.
"We need to find an effective answer to this problem."
Warning that support for EU membership in Britain is "wafer thin", Mr Cameron said: "Countries need flexibility so they can make changes to their welfare systems to better manage migration."
And he asked: "Are we going to find the flexibility to address the concerns of the UK and work together to fix this?"
The Prime Minister is under intense domestic pressure to secure substantial concessions ahead of the referendum on EU membership promised by the end of 2017.
Downing Street has said his presentation to leaders - over a dinner without any of their diplomatic entourage - was designed to put some political momentum into the renegotiation process.
Mr Cameron has already been forced to abandon his initial hope that a deal could be finalised before Christmas, and now hopes it will be possible by the next summit in February.
June is widely mooted as his favoured schedule for holding the referendum, before a summer in which Europe's migrant crisis could reach new heights.
Labour leader Jeremy Corbyn, who has been in Brussels attending a meeting of the European parliament's socialist bloc, accused Mr Cameron of pursuing irrelevant reforms because he could not manage tensions in the Conservative Party.
Mr Corbyn told the BBC: "He has come here for an argument without being very clear about what the argument is about.
"Why can't he just say, 'Well, actually, this argument really belongs within the Conservative Party', and perhaps the Conservative Party should sort out what their problem is first."
But the Labour leader also appeared to question his party's previous commitment to a two-year ban on migrants getting in-work benefits.
He reportedly told the Politico website: "If someone is working, paying taxes like anyone else, he should have access to the same benefits as everyone else."
Meanwhile, HM Revenue & Customs has come under fire for refusing to release statistics relating to UK immigration because it could prejudice the renegotiation bid.
Former Whitehall economist Jonathan Portes used freedom of information rules to request figures on how many National Insurance numbers allocated to European nationals were active after noticing a discrepancy between the number issued and official immigration estimates.
An HMRC spokesman said the response had been issued in error and the figures were expected to be published in the new year.
He said: "It was wrong to suggest information was withheld because of the EU renegotiation.
"Not all the information that has been requested is held by HMRC. The information that is held is not in a collated, publishable form.
"HMRC will release that data when it is properly collated early in the New Year."