The EU’s lead Brexit negotiator has rejected Boris Johnson’s demands for the Irish backstop to be scrapped.
Michel Barnier said the backstop – intended to avoid a hard border on the island of Ireland – was the “maximum flexibility” the EU could offer.
Mr Johnson has previously told the EU the arrangement must be ditched if a no-deal Brexit was to be avoided.
Meanwhile, the PM has told rebel Tories they face a “fundamental choice” of siding with him or Jeremy Corbyn.
His comments come as some MPs who oppose a no-deal Brexit – including Conservatives – are planning to take action in Parliament next week.
The UK is due to leave the EU on 31 October, with or without a deal.
The backstop is part of the withdrawal agreement negotiated between Brussels and former Prime Minister Theresa May, which has been rejected by Parliament three times.
If implemented, it would see Northern Ireland staying aligned to some rules of the EU single market, should the UK and the EU not agree a trade deal after Brexit.
Mr Johnson has said there has been some movement from the EU, as he attempts to broker a new deal and remove the arrangement, which he has described as “undemocratic”.
However, writing in the Sunday Telegraph, Mr Barnier said: “On the EU side, we had intense discussions with EU member states on the need to guarantee the integrity of the EU’s single market, while keeping that border fully open.
“In this sense, the backstop is the maximum amount of flexibility that the EU can offer to a non-member state.”
Mr Barnier also said he was “not optimistic” about avoiding a no-deal Brexit, but “we should all continue to work with determination”.
He added: “The EU is ready to explore all avenues that the UK government may present and that are compatible with the withdrawal agreement.”
The EU could not stop the UK from leaving without a deal, he said, but he “would fail to understand the logic of that choice” because “we would still need to solve the same problems after 31 October”.
Mr Johnson says he wants to leave the EU on 31 October with a deal, but it is “do or die” and he is willing to leave without one rather than miss the deadline.
That position has prompted a number of opposition MPs to come together to try to block a possible no deal.
MPs opposed to a no-deal Brexit are expected to try and seize control of the Parliamentary agenda this week to push through legislation that would force the PM to seek a Brexit extension beyond 31 October.
But Mr Johnson has warned Tory MPs who are considering lining up with opposition groups that they risk plunging the country into chaos.
In an interview with the Sunday Times, he said: “I just say to everybody in the country, including everyone in Parliament, the fundamental choice is this: Are you going to side with Jeremy Corbyn and those who want to cancel the referendum?
“Are you going to side with those who want to scrub the democratic verdict of the people – and plunge this country into chaos.
“Or are you going to side with those of us who want to get on, deliver the mandate of the people and focus with absolute, laser-like precision on the domestic agenda?”
His comments come after the Sun reported that No 10 would stop any Tory MP who votes to block a no-deal Brexit from standing for the party in a general election.
The report prompted former chancellor Philip Hammond to say it would be “staggeringly hypocritical” for the government to sack Conservative MPs who rebel over its Brexit plans, as eight current cabinet members had themselves defied the party whip this year by voting against Theresa May’s Brexit deal.
On Saturday, demonstrations were held across the UK in response to Mr Johnson’s plans to suspend Parliament in the run-up to Brexit.
The prime minister, who announced the move on Wednesday, said it would enable the government to bring forwards new legislation.
But the decision prompted an angry backlash from some politicians and opponents of a no-deal Brexit.
Labour leader Jeremy Corbyn accused Mr Johnson of a “smash and grab on our democracy” in order to force through no deal by leaving MPs too little time to pass laws in Parliament aimed at preventing such an outcome.
But Chancellor Sajid Javid defended the decision, adding: “It’s right because we are focusing on the people’s priorities.”
If the prorogation happens as expected, Parliament will be closed for 23 working days.
MPs have to approve recess dates, but they cannot block prorogation.