The EU will decide at the end of the week whether a Brexit deal is going to be possible, French President Emmanuel Macron has told Boris Johnson.
President Macron said talks should now proceed swiftly to see if an agreement could "respect" EU principles.
Mr Johnson said the EU should not be "lured" into thinking there would be a delay to Brexit beyond 31 October.
The PM will hold further calls with EU leaders on Monday to discuss his latest proposals for the Irish border.
Downing Street said Mr Johnson was expected to hold phone calls with the leaders of Sweden, Denmark and Poland, after speaking to Mr Macron on Sunday.
The prime minister told the French president over the phone he believed a deal could be achieved, but that the EU must match compromises made by the UK.
A French government official said President Macron told Mr Johnson "that the negotiations should continue swiftly with Michel Barnier's team in coming days, in order to evaluate at the end of the week whether a deal is possible that respects European Union principles".
The comments come ahead of a key few days of negotiations as both parties try to find a new agreement in time for a summit of European leaders on 17 and 18 October.
On Monday, Mr Johnson's Europe adviser, David Frost, will hold further discussions with the European Commission, while Brexit Secretary Stephen Barclay will visit EU capitals.
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.
A senior Number 10 source said: "The UK has made a big, important offer but it's time for the Commission to show a willingness to compromise too. If not the UK will leave with no deal."
It's good to talk.
But was there a meeting of minds between the prime minister and the French president?
Boris Johnson's aim was to disabuse President Macron of any suspicion that parliament simply wouldn't allow the UK to leave the EU at the end of the month without a deal.
So this really could be the 'final opportunity' to seal one.
The prime minister will deliver a similar message to other EU leaders.
So far, though, Mr Johnson's proposals are yet to open the door to more intensive negotiations.
From the Elysee Palace's account of the call, Macron's message to the PM seemed to be: First, work through the EU negotiator, Michel Barnier - don't work on individual leaders.
And second, if you don't move a bit more towards the EU's position by the end of the week, then it's no deal.
So far, then, any talks seem to resemble the denouement of Reservoir Dogs - more stand-off than mutual understanding.
Under the Benn Act, passed last month, the prime minister must write to the EU requesting a Brexit extension if no deal is signed off by Parliament by 19 October, unless MPs agree to a no-deal Brexit.
Government papers submitted to a Scottish court said that Mr Johnson will comply, despite his assertion that there will be "no more dither or delay".
The Number 10 source called the legislation a "surrender act" and said its authors were "undermining negotiations".
"If EU leaders are betting that it will prevent no deal, that would be a historic misunderstanding," they said.
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"
Mr Johnson has claimed his plans will be supported by Parliament.
At the weekend he 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.
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".
Speaking on the BBC's Andrew Marr Show, Brexit Secretary Stephen 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 proposals to a vote in Parliament to test support for them ahead of the EU summit.
Meanwhile, Jeremy Corbyn is set to meet the leaders of other opposition parties to scrutinise the government's new Brexit proposals.
The cross-party meeting between Labour, the SNP, Lib Dems, Greens and others on Monday will decide the next steps to "hold the government to account".
Timeline: What's happening ahead of Brexit deadline?
Monday 7 October - Opposition leaders meet to discuss the PM's border proposals. In Edinburgh, the Court of Session could rule on the sanctions Boris Johnson would face if he contravened the law compelling him to seek a Brexit delay.
Tuesday 8 October - Last working day in the House of Commons before it is due to be prorogued - suspended - ahead of a Queen's Speech to begin a new parliamentary session.
Monday 14 October - The Commons is due to return, and the government will use the Queen's Speech to set out its legislative agenda. The speech will be then be debated by MPs throughout the week.
Thursday 17 October - Crucial two-day summit of EU leaders begins in Brussels. This is the last such meeting currently scheduled before the Brexit deadline.
Saturday 19 October - Date by which the PM must ask the EU for another delay to Brexit under the Benn Act, if no Brexit deal has been approved by Parliament and they have not agreed to the UK leaving with no-deal.
Thursday 31 October - Date by which the UK is due to leave the EU, with or without a withdrawal agreement.