The decision could open the door to Parliament returning as early as next week, setting up another showdown over Brexit between Mr. Johnson and a rebel alliance of opposition and Conservative MPs.
Mr Barclay and Mr Barnier discussed three papers submitted by London with ideas on replacing the backstop provisions in the Brexit deal struck a year ago but rejected by British MPs. Using his authority as an elder statesman, Major has said Britain should remain in the European Union, and has warned that a no-deal Brexit would be disastrous for the nation.
At the end of three days of hearings, Supreme Court President Brenda Hale said the case was not about when and how Britain would leave the European Union but only about the lawfulness of Johnson's advice to the queen. It also has not ruled out trying to suspend Parliament for a second time.
Thursday also saw the final day of the legal battle over Johnson's decision to suspend Parliament at the UK's Supreme Court.
But in the closing speeches in the Supreme Court, barrister Lord Pannick said if the government loses the case, parliament should resume as soon as possible next week.
Irish Prime Minister Leo Varadkar, who will be among those at the summit, said he would try to get a deal with Johnson, reports the BBC.
The reason given by Johnson for suspending parliament for five weeks was that he needed time to prepare a new legislative agenda.
Government lawyer Richard Keen said the prime minister's opponents were "inviting the courts into forbidden territory and into what is essentially a minefield".
Major said the inescapable conclusion was that "the decision was in fact substantially motivated by a desire to obstruct Parliament from interfering with the prime minister's plans".
"I told Boris Johnson that I have no emotional attachment to the Backstop".
Major, who was prime minister from 1990 to 1997, told the court in a 10-page written submission: "The current factual picture, on the material which is available and with regard to the absence of evidence which ought to be available but has not been provided, is deeply concerning".
Finland now holds the rotating presidency of the 28-nation bloc.
Finland now holds the rotating presidency of the Council of the European Union, and the EU's other 25 Member States will have to agree to this new deadline, the original agreed as the 17th of October, the date of the European Summit.
Britain says it has not revealed detailed proposals because they would likely leak, to the detriment of negotiations.
The "confidential technical non-papers" were not the formal written legal proposals Brussels has asked for, and diplomatic tensions are mounting as the deadline approaches for London to leave the political and trading bloc it joined in 1973.
The EU confirmed it had received new documents from Britain relating to ways of maintaining an open border between EU member Ireland and the U.K.'s Northern Ireland - the key sticking point to a deal.
Earlier, Downing Street said Mr Johnson will not be bound by an "artificial deadline" to produce formal written proposals to resolve the deadlock and would submit them "when we are ready" and when Brussels is prepared to "engage constructively on them".
Britain has suggested implementing what it calls "alternative arrangements" - a mix of technology to replace border checks and a common area for agricultural products and animals covering the whole island of Ireland.
However, the Europeans have talked tough before, signalling their preparedness to cut the United Kingdom off from the EU, before granting one Brexit extension in April and then another to October.
Barclay insisted that the six weeks until October 31 were "sufficient for a deal" if both sides provided "creative and flexible solutions". The document tweeted by Maugham was their response, he said. "But we are making progress".
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