The ability of the British PM to deliver a plan that will pass through the House of Commons arose after Dominic Raab, the Brexit secretary, informed the EU's chief negotiator, Michel Barnier, on Sunday that May could not get an agreement past her cabinet or the DUP, on whose votes her government heavily relies.
Mrs May has suggested delaying the UK's final departure from the EU's until 2021 in a last ditch bid to end the deadlock over the border issue.
Arriving for the second day of the European Council summit, Mrs May made clear she would accept an extension only as a means to ensure there was no hard border in Ireland if it proved impossible to implement the future partnership by the end of 2020.
"I am open to the idea of an extension or a longer transition period, but that's not an alternative to a legally binding Irish backstop, so it's certainly not an alternative; it's something that might be part of the mix", he said.
"The Gibraltar protocol is resolved, it's been closed with the British government, and what we are doing now is working out the four memorandums for the bilateral relationship between the government of Spain and the United Kingdom", said Sánchez.
The possibility of extending the transition period, set to end by December 2020, was also mooted at the Brussels summit.
Former Ukip leader Nigel Farage said any extension to the transition period could delay full withdrawal nearly until the general election scheduled for May 2022, and "may mean we never leave at all".
The DUP's support is crucial for the continuation of May's government after her party lost the majority in parliament, following a snap election a year ago.
Speaking at a news conference on Thursday, Tajani said, "As far as we are concerned, this is key".
The former Brexit minister has put down amendments to the Northern Ireland bill to that effect.
The future U.K. -EU border in Ireland has been the main sticking point in the talks.
She said the British Government was incapable of acting responsibly or with any semblance of impartiality while it remained wedded to the DUP.
European Commission President Jean-Claude Juncker backed the plan as a "good idea", claiming "this prolongation of the transition period probably will happen".
"That is not an acceptable way for a leader of a government to behave". But she said the extra time was merely an insurance policy and was unlikely to be needed. More like kicking the can further down the road and delaying, by a bit, driving off the cliff. But even an extension would not get rid of the EU's insistence that such a backstop must be agreed to secure a deal.
"Despite this, sterling continues to keep calm and carry on despite the obvious Brexit threat with negotiations on this front clearly not progressing as many would've hoped".
"We are close to the moment of truth".
A senior European Union official later said Mrs May had indicated she was "ready to consider" a longer transition period.
"I believe we need a deal".
But she later accepted a 21-month transition offered by the European Union, ending on the last day of December 2020.
However, Brexiteers have reacted with anger at suggestions the UK's stay within the EU's structures could be lengthened beyond what had previously been agreed as a 21-month period.
He also warned that a hard border between Ireland and Northern Ireland could lead to violence.
The two sides remain deadlocked, and this week's summit, which had been billed as a make-or-break moment, turned simply into a chance for Britain and the European Union to give themselves more time - perhaps until the end of the year - to break the logjam.