Germany has proposed to EU states that they should agree to drop the idea of a constitution when they meet at a summit in Brussels on Thursday and Friday.
Angela Merkel faces determined opposition from Poland
The "constitutional concept... is abandoned", says a paper circulated by Germany, which will chair the summit.
The paper makes several concessions to EU member states opposed to key parts of the failed constitution.
But Poland and the UK are still warning they could use their vetoes if they do not get their way on a new treaty.
Polish Prime Minister Jaroslaw Kaczynski said a veto remained "highly likely" while UK Foreign Secretary Margaret Beckett said no deal was better than a bad deal.
It does seem the British government has largely got its own way, although there's a lot of detail to be settled
Mrs Beckett told Parliament: "If it comes down to deal or no deal at this European Council the UK government is clear... no deal is better than buying any old pig in a poke."
She added that Britain wanted a Europe "of sovereign nations, not a superstate".
The summit is intended to issue a mandate for an intergovernmental conference to agree the precise wording of a treaty to replace the failed constitution.
If it fails, it will plunge the EU into a political crisis as deep as the one that followed the rejection of the constitution by French and Dutch voters two years ago.
The German paper proposes that the new treaty is called "The Reform Treaty", accepts that the EU will not have a "foreign minister", and provides countries with a chance to opt out of EU policies in the area of policing and criminal law.
Correspondents say the biggest remaining problem for the UK could be Germany's continued support for the idea of making the EU's Charter of Fundamental Rights legally binding.
The UK fears this could allow the European Court to make decisions that would change British labour law.
The part of the constitution that Poland most disliked - the introduction of a new voting system for decisions taken by member states - is preserved in the latest German proposals, seen by the BBC.
However, the BBC's Oana Lungescu in Brussels says the German paper offers Poland a "discreet olive branch".
For the first time it mentions, in a footnote, that the Poles, backed by the Czechs, want to raise the idea of changing the voting system at the summit.
The new system would benefit larger member states to the detriment of smaller and medium-sized ones, and have the effect of reducing Poland's clout.
Despite his warnings of a possible veto, Mr Kaczynski hinted Poland could drop its opposition if it guaranteed a strong voice in EU decision-making.
"We realise we cannot stop the process [of reform] - that would be too risky for the future," Mr Kaczynski told Reuters news agency.
He said there was a 50-50 chance of the summit ending in agreement.
Under the latest German proposals, Britain gets reassurances that the European Courts will have no power to examine foreign affairs policies.
And at the request of the Dutch, the draft gives more power to national parliaments to block EU laws.
But a Dutch proposal to enshrine criteria for further enlargement in the treaty, is not fully satisfied.
That would send a very negative signal to Balkan countries, an EU diplomat said.