EU sources say the European chief Brexit negotiator, Michel Barnier, has repeatedly warned the “clock is ticking” but it is not lack of time that stands in the way of being struck.
EU officials and diplomats have said there needs to be a fundamental change in UK politics for there to be any value for the EU in an extension of British membership.
One diplomat said: “We want to de-dramatise this, talk it all down, and so there was surprise at the aggressive tone.
“It just gives her even less room for manoeuvre. If she wants to give up, she should just say so.”
Article 50 can only be extended after unanimous agreement of the 27 other EU member states and on the request of the UK.
But Theresa May has already said she is not prepared to slow down the Brexit timetable.
EU officials and diplomats do not see any purpose in an extension to article 50 unless there is a change in politics.
One senior diplomat involved in the negotiations said: “If it is time just for the sake of putting off an inevitable no deal, then it will not happen.”
Speaking to The Guardian, director of the Centre for European Reform thinktank, Charles Grant, said: “If there is a very good reason for it, then fine.
“But if it is just because a deal hasn’t been agreed, then that’s not going to be fine.
“The Commission and the French government say they are relaxed about a no deal, although I am not sure they are.”
Although extending article 50 has not been officially debated, informal discussions suggest there is little support among member states, sources in Brussels claim.
One diplomat involved in Brexit negotiations said: “You have 27 member states who have to agree to it and I can’t see them all doing so.
“There will always be someone who is awkward, and wants something out of it. Spain might want to make a point about Gibraltar, for example.
“We have spent the GDP of Greece in terms of manpower in trying to first of all keep the Brits in and now negotiate on their exit.
“Even if the UK asked for it, and that seems unlikely, it would need a compelling reason to be given.”
Former Treasury and European Commission official, Mujtaba Rahman, said the EU parliament is against an extension as it would play well for populists, including UKIP.
A source in the parliament said: “This has been repeatedly discussed.
“It would be the perfect opportunity for UKIP to rebuild, which is what no one wants.
“After all, Guy Verhofstadt has said before that the only good thing about Brexit is that Nigel Farage won’t be coming to Brussels again.
“The parliament won’t want them back again”.