Boris Johnson has claimed his Brexit proposals have picked up support in Parliament, as he urged the EU to compromise.
Ahead of a crucial summit in 11 days' time, he insisted a revised agreement was possible "if the EU is willing."
Latvian PM Krisjanis Karins said a new deal "may be a little bit of a long-shot" but was "certainly possible".
Brexit Secretary Stephen Barclay urged the EU to intensify negotiations on the UK's plans.
Talks are due to resume on Monday as both parties try to find a new agreement in time for the summit of European leaders on 17 and 18 October.
But arrangements for preventing a hard border on the island of Ireland continue to be a sticking point, with the EU calling for "fundamental changes" to the UK's latest proposals.
Speaking on the BBC's Andrew Marr Show, Mr Barclay said more advanced negotiations would need to begin "in the coming days" for a deal to be reached before the current Brexit deadline of 31 October.
Irish Prime Minister Leo Varadkar said he believed a deal was possible but said current proposals did not go far enough.
French officials said that in a telephone conversation with Mr Johnson on Sunday, French President Emmanuel Macron urged him to swiftly continue talks on the UK's new plans for the Irish border with European Commission officials.
Mr Barclay said talks were under way with Labour and other opposition MPs aimed at securing their support for a new deal.
He said ministers were "considering" the idea of putting the PM's new proposals to a vote in Parliament to test support for them ahead of the EU summit in mid-October.
But speaking on the same programme, Shadow Attorney General Shami Chakrabarti said the government would have to "compromise" to get Labour's support for a new agreement, and a deal would be "more likely" to meet the party's tests if it won the support of Ireland.
"The deal that he's currently proposing - that is not going down very well in Brussels, or in Dublin or with us - is a deregulatory deal that business in this country doesn't want," she said.
The UK has said there is "no path" to a deal without a replacement for the Irish border backstop plan, which is opposed by many MPs.
The backstop is the controversial insurance policy that is meant to keep a free-flowing border on the island of Ireland but which critics - including the PM - fear could trap the UK in EU trading rules indefinitely.
"I say to our European friends: grasp the opportunity our new proposal provides. Join us at the negotiating table in a spirit of compromise and co-operation," he said.
He claimed MPs from "every wing of the Conservative Party", Northern Ireland's Democratic Unionist Party and from Labour have said "our proposed deal looks like one they can get behind".
But he said "there will be no more dither and delay" and the UK would leave the EU on 31 October with or without a deal.
Mr Johnson did not explain how the government would comply with a law passed by MPs which forces the prime minister to seek an extension to the Brexit deadline if no agreement has been made by 19 October.
In court documents, the government has said the prime minister will request a delay as the law requires, despite his public and Parliamentary statements.
What are the PM's border plans?
Under Mr Johnson's proposals, which he calls a "broad landing zone" for a new deal with the EU:
- Northern Ireland would leave the EU's customs union alongside the rest of the UK, at the start of 2021
- But Northern Ireland would continue to apply EU legislation relating to agricultural and other products, if the Northern Ireland Assembly approves
- This arrangement could, in theory, continue indefinitely, but the consent of Northern Ireland's politicians would have to be sought every four years
- Customs checks on goods traded between the UK and EU would be "decentralised", with paperwork submitted electronically and only a "very small number" of physical checks
- These checks should take place away from the border itself, at business premises or at "other points in the supply chain"
Baroness Chakrabarti said the prime minister "speaks with forked tongue" on the possibility of asking for an extension, adding he "seems to think he is above the law".
She insisted there were no loopholes in the legislation, adding that the conditions under which the PM must ask for another Brexit delay were "explicit and specific".
Mr Barclay confirmed he would be travelling to the Netherlands on Sunday evening as part of an effort to secure support for the PM's plan in Europe, and urged the EU to show "creativity and flexibility".
He indicated the UK was open to some flexibility on how authorisation from Northern Irish politicians would have to be secured for parts of the PM's plan to come into effect.
"The key issue is the principle of consent, that's why the backstop was rejected three times," he said.
"So the key is the principle of consent, now of course in the mechanism, as part of the intensive negotiations we could look at that and discuss that."
Latvian Prime Minister Krisjanis Karins told the Andrew Marr Show the UK's border ideas were a "basis for negotiations", although he noted the EU "doesn't have a whole lot of wriggle room".
But he said: "It depends on one crucial element: that also Mr Johnson as well as the EU are willing and ready to move in a compromise manner. If the offer from the UK turns out to be 'take it or leave it', it's going to be very difficult."
Mr Varadkar said on Saturday a deal was "still possible" but the EU did not believe the current proposals from Mr Johnson "form the basis for deeper negotiations".
The EU is concerned the UK wants to leave too many details about customs and regulatory checks to be agreed during the transition period after Brexit.