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 can "respect" EU principles.
Mr Johnson said the EU should not be "lured" into thinking there will be a delay to Brexit beyond 31 October.
However, a law requires him to request one if a deal is not agreed by 19 October.
As part of a weekend talking to EU leaders, the prime minister told President Macron over the phone he believes a deal can 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."
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.