Brexit negotiations are continuing on a final day of efforts to get a deal ready for a crucial EU summit, after Tuesday’s talks ran into the small hours of the morning.

Boris Johnson is running out of time to get an agreement in place so it can be approved by European leaders at the Brussels summit starting on Thursday.

A Number 10 source said progress was still being made in the talks, which ran to about 1.30am in the Belgian capital and resumed on Wednesday morning.

Reports had suggested a deal was close ahead of a midnight deadline imposed by the EU, with the Prime Minister said to be making major concessions on the Irish border.

But sources on both sides of the Channel downplayed the suggestions, and the PM’s official spokesman said: “Talks remain constructive but there is more work still to do.”

The PM is expected to update his Cabinet on progress in the negotiations on Wednesday afternoon.

A Downing Street source said: “Constructive talks, worked into the night, continue to make progress, continue in the morning.”

If the PM succeeded in bringing a deal home to the UK, he would then face a battle to do what Theresa May failed to do three times and get it approved by Parliament.

Hardline Brexiteers from the European Research Group (ERG) of Tory backbenchers and the DUP were both courted in Downing Street on Tuesday.

ERG chairman and self-styled “Brexit hardman” Steve Baker later said he was “optimistic” that Mr Johnson’s team would finalise a “tolerable deal that I will be able to vote for”.

But a split among the hardliners became visible over reports that Mr Johnson may agree to a border down the Irish Sea.

Former environment secretary Owen Paterson said in an interview with The Sun that it was “unacceptable” that custom checks could be carried out on goods travelling between Northern Ireland and Great Britain.

Jacob Rees-Mogg, the Commons leader and a former ERG chair, however thinks an agreement could be accepted by MPs, telling LBC: “I think the votes are there now for a deal.”

The DUP has helped prop up the Tory administration following increased funding from Mr Johnson’s predecessor in Number 10 as part of a confidence and supply deal, in a £1 billion package branded as a “bribe”.

There is speculation more money could be headed in the way of Arlene Foster’s party as the PM tries to get them on board with any concessions.

The Guardian reported senior sources on both sides of the Channel saying that a draft treaty could be published on Wednesday morning after the UK agreed in principle there will be a customs border in the Irish Sea.

The DUP were decidedly lukewarm on the mooted proposals in a statement after their second audience with the PM in as many days.

“We respect the fact negotiations are ongoing and therefore cannot give a detailed commentary but it would be fair to indicate gaps remain and further work is required,” a spokesman said.

Former Brexit secretary David Davis said DUP backing would be influential for his colleagues.

“There will be quote a lot of Tory MPs who will take their line from what the DUP do,” he told BBC Radio 4’s Today programme.

He was reminded that a customs border in the Irish Sea was once described by DUP leader Arlene Foster as a “blood red” line.

“Well let’s see when she sees the detail of the deal whether she thinks this is a blood red line or an acceptable compromise,” Mr Davis said.

Pressure to sign off on a draft agreement is peaking. A legal text needs to be published ahead of the summit if the EU27 are to consider ratifying the Withdrawal Agreement at the two-day summit.

Their approval would allow Mr Johnson to put the deal to MPs in a proposed extraordinary sitting of Parliament on Saturday.

This would give him a chance to avoid a monumental clash over asking for a Brexit extension – something he has repeatedly ruled out doing but is compelled to do so under the Benn Act.

That legislation passed by MPs opposed to a no-deal, including Tory rebels, orders the PM to ask Brussels for an Article 50 extension until the end of January if Parliament does not back a deal.

But there are fears Mr Johnson will find a loophole to avoid making the request as part of his “do or die” commitment to leave the bloc by the October 31 deadline.

If a Saturday showdown in Parliament is to take place the Government must put a motion before the Commons on Wednesday, to be voted upon on Thursday.

During the weekend session, MPs would be able to back or reject any deal presented to them, or discuss what to do next in the Brexit saga.

The Liberal Democrats have put forward an amendment to the Queen’s Speech for Tuesday to test whether there is support for a second referendum.

And leader Jo Swinson has suggested her Remain-backing party could support a Johnson deal if it is put back to the people in a confirmatory vote.

“We will back a referendum whether it’s on Boris Johnson’s deal, whether it’s on Theresa May’s deal, because we think it should be the public that are in charge and are having that say,” she told Today.

Sam Blewett is PA Political Correspondent.