Theresa May has been given 10 days to improve her offer to the EU if she wants its leaders to allow Brexit talks to move on to transition and trade.

After a meeting in Brussels, European Council President Donald Tusk said Ms May would have to finalise her deal by 4 December when she is expected to meet other key EU officials.

If Ms May’s offer on critical withdrawal issues including the “divorce bill”, EU citizens’ rights and the Irish border meets with approval, it will allow EU leaders to rule that “significant progress” has been made for negotiations to move forward at the European Council on 14 December.

The Prime Minister hopes the crunch summit in Brussels will give a boost to the whole Brexit process and to the credibility of her leadership, which hinges on making a success of EU withdrawal.

Friday saw the issue of the Irish border come under intense focus, amid increasing pressure for better guarantees on the issue from Dublin.

After seeing Ms May on Friday, Mr Tusk said it was “possible” sufficient progress could be made at the December summit but that it remained a “huge challenge”.

He added: “We need to see progress from UK within 10 days on all issues, including on Ireland.”

Brexit: the deciders Show all 8 1 /8 Brexit: the deciders Brexit: the deciders European Union's chief Brexit negotiator, Michel Barnier Getty Brexit: the deciders French President Emmanuel Macron Getty Brexit: the deciders German Chancellor Angela Merkel Reuters Brexit: the deciders Commission President Jean-Claude Juncker EPA Brexit: the deciders The European Parliament's chief Brexit negotiator Guy Verhofstadt Getty Brexit: the deciders Britain's Prime Minister Theresa May Getty Images Brexit: the deciders Britain's Chancellor of the Exchequer, Philip Hammond PA Brexit: the deciders After the first and second appointed Brexit secretaries resigned (David Davis and Dominic Raab respectively), Stephen Barclay is currently heading up the position PA

As she left the gathering of European leaders, Ms May said: “There are still issues across the various matters we are negotiating on to be resolved, but there has been a very positive atmosphere in the talks and a genuine feeling that we want to move forward together.”

Taoiseach Leo Varadkar has warned that the deadlock in Brexit negotiations cannot be broken until the border issue is resolved, and has demanded fresh assurances there will be no hard border between Northern Ireland and the Republic.

Irish foreign minister Simon Coveney said on Friday it was difficult to see how border checks could be avoided if the UK quit the EU’s customs union and single market, leading to “regulatory divergence” between the North and the Republic.

Mr Varadkar has previously suggested a “bespoke” arrangement, similar to that operated on the Isle of Man, under which Northern Ireland, or the whole of the UK, would continue to observe the rules of the single market and customs union without necessarily remaining a member of them.

Meanwhile, the EU’s chief Brexit negotiator Michel Barnier discussed the situation with Mr Coveney in Brussels, saying afterwards that there is “strong solidarity” with Ireland and adding that “Irish issues are EU issues”.

Ms May responded: “In relation to the border between Northern Ireland and the Republic of Ireland, we and the Irish government continue to talk about the solution to that.

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“But we have the same desire – we want to see that movement of people and trade across that border can carry on as now and that we don’t create any new barriers to trade or the movement of people across that border.”

But back in London, Downing Street wobbled over suggestions that Northern Ireland’s continued membership of the EU customs union could be up for negotiation in Brexit talks.

A No 10 spokesman told reporters on Friday that the UK continued to look for “an innovative way forward” on the issue.

Irish leader Leo Varadker wants stronger assurances on the Irish border issue (AFP/Getty)

Asked whether Northern Ireland could remain in the customs union following Brexit, the spokesman said: “That is a matter for negotiations.”

But a Downing Street source later insisted the Government’s position is still that the whole of the UK will leave both the customs union and single market after Brexit.

The confusion underlines the political difficulty of Ms May’s position, with the Northern Irish DUP propping up her Commons majority and dead set against anything that would see Belfast and London diverge.

As well as talks with Mr Tusk and Ms Merkel in Brussels, the Prime Minister had meetings with Danish premier Lars Lokke Rasmussen, Belgium’s Charles Michel, and Lithuania’s Saulius Skvernelis in the margins of the Eastern Partnership summit.

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One of the issues under discussion was thought to be the amount the UK is prepared to offer Brussels following reports that cabinet ministers agreed to double the sum originally put on the table by Ms May to around £40bn.

The Prime Minister does not want to name a precise figure until she has a clear idea of what kind of trade deal is available with the remaining EU member states in the phase two negotiations.

She said: “We have been talking about how we can progress the issue in relation to the financial settlement. I have set out the position. I did so in the Florence speech.