THERESA May today begins a major diplomatic push to seal a Brexit deal in time for the European Council summit in two weeks’ time.

The Prime Minister will hit the Downing Street phones in calls to her European counterparts in a frantic bid to square the circle on a withdrawal agreement as the EU27 ups its own diplomatic tempo with Ireland’s Leo Varadkar arriving in the Belgian capital for talks with Donald Tusk, the European Council President, and Michel Barnier, the European Commission's chief negotiator.

The Taoiseach is also due to meet Guy Verhofstadt, who chairs the European Parliament's Brexit Steering Group.

On Wednesday, the EU27 announced that its leaders would hold a private dinner the evening before the October 18/19 Brexit summit.

The impasse over the Irish border remains the key sticking point with the UK and EU unable to find common ground on the shape of a legally binding “backstop” position that would ensure free movement across the frontier even if a wider trade deal failed to materialise.

Mrs May has admitted she is preparing to put forward changes to her controversial Chequers Plan and one option she is said to be working on would be meeting Brussels “half-way” on the Irish backstop should there be no long-term agreement on how to move forward on the border issue.

The first part of the compromise would involve Brussels accepting the whole of the UK taking part in a de facto customs union with the EU, thus maintaining the integrity of the Union.

It would be a “temporary” extension of the transition period until a full trade deal could be agreed, although some officials believe this might run on for years beyond December 2020.

While this has already been rejected by Mr Barnier because it smacks of cakeism ie the UK would benefit from being part of the EU’s regulatory regime but would have none of the obligations a member has, Dublin is said to be favourable to it because it would maintain frictionless trade across the Irish border.

The second part of the compromise would meet the EU’s desire that Northern Ireland would stay part of the single market regulatory area of the bloc at least on matters of food and animal welfare.

The leadership of the Democratic Unionist Party has come out against any regulatory checks at Irish Sea ports and its leader Arlene Foster has warned that her party’s red lines are “blood red”. But it is thought that most of the regulatory checks would be done at company premises or via a trusted-trade scheme.

Tomorrow, Mr Barnier is due to meet politicians from Northern Ireland in Brussels. Dominic Raab, the Brexit Secretary, is expected to travel to the Belgian capital in the next few days to present the Government’s new Chequers Plan.

Politically, any further compromise on Mrs May’s part is dangerous. The tweaked Chequers proposal will not find favour with Tory Brexiteers, who will see it as a further betrayal. Nor is it likely that the DUP would accept a customs fudge however temporary it may or may not be.

Yet whatever deal the PM brokers with Brussels has to get through Westminster, where the majority leans to a softer Brexit approach. If Mrs May is to succeed, it could be that in the final vote, and with MPs faced with her deal agreed with the EU27 or no-deal, she has to rely on Labour Remainers, defying Jeremy Corbyn, to push her agreement through.

Despite all the difficulties, there appears to be real optimism in Brussels that a Brexit deal can be done by the October 18/19 summit. One senior EU diplomat said: “It will be long hours and it may get messy but there will be a deal. The cost of no-deal is just too big.”