Relaying comments made by the EU's Brexit negotiator, Michel Barnier, at a meeting of the bloc's ministers in Luxembourg, they said that if the Tuesday deadline was missed then talks would have to continue after the Thursday-Friday summit.
"We need to land this tonight," an EU official told Reuters.
However, British Brexit Minister Stephen Barclay struck a more positive note as he arrived in Luxembourg for talks with ministers from the 27 countries that will remain in the EU.
"The talks are ongoing we need to give them space to proceed," he told reporters. "Detailed conversations are underway and a deal is still very possible."
The main sticking point to reaching a deal before the October 31 scheduled date for Britain's exit is customs and security arrangements for the border between EU member Ireland and the British province of Northern Ireland - the only land frontier between the EU and the United Kingdom after Brexit.
Barnier said earlier that while an agreement was still possible this week, it was still possible and it was "high time to turn good intentions in a legal text".
Finland's EU affairs minister, Tytti Tuppurainen, said the EU must prepare for a no-deal and a third extension of the tortuous divorce process.
"All scenarios are open," she said, adding that leaders at the summit would discuss pushing back the departure date beyond October 31.
British Prime Minister Boris Johnson, a leading figure in the 2016 referendum in which Britons opted to leave the EU, has pledged to take the United Kingdom out of the bloc on that day whether or not a withdrawal agreement has been reached.
But the British parliament has passed a law saying the country cannot leave without an agreement, and Johnson has not explained how he can get around that.
Malta's EU minister, asked if he would bet on there being a Brexit deal this month, said: "I would not."
Watch: Make-or-break week for Brexit deal
Deal, No-Deal, Delay
As ministers met in Luxembourg, British Brexit negotiator David Frost was starting another round of negotiations with the EU's executive European Commission in Brussels. Britain was due to make new proposals in an attempt to break the Brexit deadlock, RTE said.
There was no comment from Johnson's office and the British mission to the EU declined to comment.
A British government source said separately that new texts have been shared repeatedly with Brussels during talks.
The main obstacle is around customs, with London proposing that Northern Ireland stays in the UK customs area but that EU tariffs are applied on all goods crossing from mainland Britain to the island.
Businesses would be eligible for a refund if it turns out their goods stay in Northern Ireland under the plan, which the EU has many doubts about, saying the system is too complicated, untested and not detailed enough at this stage.
The EU believes the only possible deal to be had at the summit is a return to a solution - already rejected by Britain previously - of keeping Northern Ireland in the bloc's customs union.
It would require a major shift from London and Johnson will face his factious parliament at a rare Saturday session after the summit.
Deal or no deal, the EU believes another delay to Britain's October 31 departure date will be needed. Extension options range from as short as an extra month to half a year or longer. The other EU states would need to agree unanimously to grant it.
"Time alone is not a solution. However, if a significant political change takes place in the UK then that could potentially justify a discussion on an extension if we were asked for it," France's Amelie de Montchalin said.
"A significant political change is the prospect of an election, a referendum, something that changes the political dynamic so that the triangle between the government, parliament and the British people could align with each other a bit more."
Should no final resolutions come at the summit later this week, the EU is mulling an extraordinary one in late October to approve any deal or a delay or give marching orders for a damaging no-deal split.