"But, the key with any delay is we are able to terminate it once we ratify in order that we can then get on and get a deal through Parliament, ratify that agreement and leave the European Union", he said.
So far, UK MPs have rejected the withdrawal agreement May reached with other European leaders in 2018, so she is now asking for the leaving date to be extended.
Discussions in Brussels are to focus on the length of the delay - the French source said a 12-month extension "seems too long" - and arrangements to limit Britain's influence within the EU. "That is why the request today is to June 30 in order that we can leave as soon as possible", he told the BBC Radio 4 Today programme.
He said: "It is Parliament that is now refusing to actually honour the referendum result, with the Prime Minister's deal voted against and the Withdrawal Agreement blocked then the only alternative is to go back to the European Union and to agree the shortest possible extension".
Mrs May has already requested a further extension of the Brexit deadline to June 30, despite European Council president Donald suggesting that EU leaders should grant the United Kingdom a longer extension of up to one year.
The Kantar poll of 1,172 people found that 51% of Britons would like a referendum to be held on the deal once negotiations between the Government and the European Union have come to an end.
Mr Barclay said: "That is a outcome of Parliament, not the Government".
He said: "Please, put us out of our misery now, as this House and this government appear incapable of doing".More news: Scientists set to reveal first-ever image of a black hole
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"This is why I believe we should also discuss an alternative, longer extension".
"One possibility would be a flexible extension, which would last only as long as necessary as and no longer than one year", he said.
The current incumbents of the roles - Mr Juncker and Donald Tusk - both favour a lengthy Brexit delay until March 2020 to help Westminster find a way to ratify the withdrawal agreement.
"The UK would be free to leave whenever it is ready".
A draft summit communique, obtained by the Guardian, and to be agreed by the EU27's leaders on Wednesday, assumes in return for an extension a "commitment by the United Kingdom to act in a constructive and responsible manner throughout this unique period in accordance with the duty of honest cooperation".
Seeking support from European heavyweights Tuesday, May flew to Berlin and Paris to plead for good terms with German Chancellor Angela Merkel and President Emmanuel Macron, who has been taking a tough stance ahead of the Brussels summit.
May arrives in Brussels for a pre-summit meeting with Tusk before explaining her delay request to the leaders who will then discuss their next move.
"We have yet to see the clear shift in the government's position that is needed to secure a compromise agreement".