The government has delivered its new Brexit proposals to the EU, including plans to replace the Irish backstop.
The plan, outlined in a seven-page document, would see Northern Ireland stay in the European single market for goods, but leave the customs union - resulting in new customs checks.
The Northern Ireland Assembly would get to approve the arrangements first and vote every four years on keeping them.
The European Commission said there had been progress but "problems" remained.
Irish Prime Minister Leo Varadkar said the new blueprint did not "fully meet the agreed objectives of the backstop", in terms of upholding the EU's single market, protecting peace in Northern Ireland and supporting economic co-operation with the Republic of Ireland.
But he said he wanted an agreement and talks would continue.
The UK is set to leave the EU on 31 October and the government has insisted it will not negotiate a further delay beyond the Halloween deadline.
Speaking at the Conservative Party conference earlier on Wednesday, Boris Johnson said the only alternative to his Brexit plan was no-deal.
In a letter to European Commission's president, Jean-Claude Juncker, the prime minister said the new proposals "respect the decision taken by the people of the UK to leave the EU, while dealing pragmatically with that decision's consequences in Northern Ireland and in Ireland".
Government sources hoped the UK might be able to enter an intense 10-day period of negotiations with the EU almost immediately, with the aim of coming to a final agreement at an EU summit on 17 October.
Mr Juncker welcomed what he said were "positive advances" in some areas but he said the UK's proposed system of "governance" of the new arrangements was "problematic" - and customs rules remained a concern.