The UK has yet to come forward with a solution to replace the backstop, says the European Commission.
Downing Street said a meeting between Boris Johnson and Commission president Jean-Claude Juncker was “constructive”.
But in a statement after the working lunch, the Commission said no proposals had been put forward to replace the controversial Brexit policy.
Mr Johnson has called it “undemocratic” and said it needs to be removed from any deal he would do with the EU.
The bloc has said it is willing to look at alternatives, but that an insurance policy like the backstop must be in place.
The backstop is a part of the existing withdrawal agreement to prevent a hard border returning to Ireland.
Both Mr Johnson and Mr Juncker – who met for the first time since the PM took office in July – agreed the discussions between the UK and EU “needed to intensify” and meetings “would soon take place on a daily basis”.
But regardless of the outcome, No 10 said the PM “would not request an extension and would take the UK out of the EU on 31 October”.
The pair were joined by the EU’s chief Brexit negotiator Michel Barnier and Brexit Secretary Steve Barclay at the meeting in Luxembourg.
The statement from the European Commission said the purpose of the meeting was to “take stock” of ongoing talks, and look to the “next steps”.
But it added: “President Juncker recalled that it is the UK’s responsibility to come forward with legally operational solutions that are compatible with the withdrawal agreement.
“President Juncker underlined the Commission’s continued willingness and openness to examine whether such proposals meet the objectives of the backstop.
“Such proposals have not yet been made.”
The statement said the Commission would “remain available to work 24/7” and said the upcoming European Council summit in October will be “an important milestone in the process”.
It concluded: “The EU27 remain united.”
The No 10 statement said Mr Johnson confirmed his commitment to the Good Friday Agreement – the peace deal brokered in Northern Ireland – and still had a “determination to reach a deal with the backstop removed, that UK parliamentarians could support”.