European Union leader Donald Tusk said that with any agreement far from being clinched and businesses fearing a chaotic and costly cliff-edge departure, keeping the Brexit day at March 29 would be too risky, AP reported.
"I believe that in the situation we are in, an extension would be a rational solution," Tusk told reporters.
He said "all the 27 (member states) will show maximum understanding and goodwill" to make possible such a postponement, which needs unanimity.
British Prime Minister Theresa May, however, immediately dug in her heels and said she could deliver on the set date, however massive the challenge.
"It is within our grasp to leave with a deal on 29th of March and I think that that is where all of our energies should be focused," May said.
She said that "any delay is a delay. It doesn't address the issue. It doesn't resolve the issue."
May met EU leaders over two days at the EU-Arab League summit in the Egyptian Red Sea resort of Sharm El-Sheikh.
May met with German Chancellor Angela Merkel and European Commission President Jean-Claude Juncker early Monday as she sought elusive changes to the UK-EU divorce agreement.
Britain's Parliament has rejected the deal once, and May has just over a month to get it approved by lawmakers before the UK's scheduled departure day.
May says a new vote won't be held this week and could come as late as March 12.
Tusk, the European Council president, said that such a timeframe might get too tight to avoid a chaotic departure.
Tusk refused to say how long such an extension should be as rumors swirled it should go to anything from two months to almost two years.
British lawmakers' objections to the Brexit deal center on a provision for the border between the UK's Northern Ireland and EU member Ireland. The mechanism, known as the backstop, is a safeguard that would keep the UK in a customs union with the EU to remove the need for checks along the Irish border until a permanent new trading relationship is in place.
May wants to change the deal to reassure British lawmakers that the backstop would only apply temporarily.
But EU leaders insist that the legally binding Brexit withdrawal agreement, which took a year and a half to negotiate, can't be reopened.
A group of British lawmakers will try this week to force the government to delay Brexit rather than see the country crash out of the bloc without a deal. They want Parliament to vote Wednesday to extend the negotiating process.
Labour lawmaker Yvette Cooper, one of those behind the move, said it was irresponsible of the government that just a few weeks before Brexit "we still don't know what kind of Brexit we are going to have and we're not even going to have a vote on it until two weeks before that final deadline."
"I don't see how businesses can plan, I don't see how public services can plan and I think it's just deeply damaging," Cooper told the BBC.