Boris Johnson has claimed his Brexit proposals have picked up support in Parliament as he urged the European Union to compromise.
Writing in two Sunday papers he said: "We are leaving in 25 days. We can do it with a deal if the EU is willing."
Meanwhile, Brexit Secretary Stephen Barclay told the EU and opposition MPs: "We are not backing down."
The UK offer must not be "take it or leave it", Latvia's Prime Minister Krisjanis Karins told the BBC.
Talks are due to resume on Monday as both parties try to find a new agreement in time for a crucial summit of European leaders on 17-18 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.
Ireland's taoiseach Leo Varadkar said he believes a deal is possible but said current proposals do not go far enough.
The UK has said there is "no path" to a deal without a replacement for the Irish backstop, which is opposed by many MPs.
Writing in the Sun on Sunday and the Sunday Express, Mr Johnson said his untested plan to use technology to eliminate customs border checks would take the UK out of EU trade rules while respecting the Northern Ireland peace process.
"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 will 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.
Speaking to the BBC's Andrew Marr Show, Latvian Prime Minister Krisjanis Karins said a deal is "certainly possible".
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 that "a deal is still possible" but the EU does not believe the current proposals from Mr Johnson "form the basis for deeper negotiations".
The EU is concerned that the UK wants to leave too many details about customs and regulatory checks to be agreed during the transition period after Brexit.
Writing in the Sunday Telegraph, Mr Barclay also said there were "positive sounds" of support from MPs but "we need that noise to grow louder" as the EU summit approaches.
But he said the UK's negotiation had been "severely hampered" by "hardcore opposition MPs", who voted to prevent a no-deal Brexit.
"My message to the EU and anti-Brexit opposition is this: we are not backing down," he said.