The government and the EU could be persuaded to extend the Brexit process if Theresa May’s deal is voted down by MPs, Labour has argued, as ministers begin the task of trying to sell the agreement to the country.

Keir Starmer, the shadow Brexit secretary, said May had “run down the clock” with her Brexit talks, meaning that to renegotiate a better permanent deal would probably require article 50 to be extended beyond March next year.

If the only alternative was the chaos of no deal, then May and the EU27 would back down and allow this, Starmer argued.

However, the Brexit secretary, Stephen Barclay, who was sent out to extol the merits of the deal agreed in Brussels on Sunday, said that if MPs voted down the deal “we go into uncharted waters”.

“This is the best deal, and it’s also the only deal, and EU leaders have made that clear,” he said.

May was expected to make the same point in a statement to parliament later on Monday, warning MPs that failing to back the deal would take Britain “back to square one”.

Barclay conceded it would be difficult to get the deal through parliament, but said the cabinet needed to make the case, for it.

“I don’t pretend for a minute it’s not a challenging task given where the numbers currently look,” he told Sky News. “But this is a good deal that respects the referendum result.

“And we need to be clear with parliamentary colleagues as to what the alternative will be, which will be massive uncertainty from either no deal or no Brexit. That’s not in the interests of their constituents or in the interests of protecting jobs. It gives uncertainty to EU citizens in areas like the NHS.”

Starmer dismissed that forecast, saying Labour’s main concern was with the vagueness of the agreement on a future relationship.

“It is a problem, and it’s a problem the prime minister has really got us into, because over the two years she has effectively run down the clock,” he told BBC Radio 4’s Today programme.

“We had a snap general election at the beginning of the exercise, which wasted the first few months. We had months and months and months of waiting for Chequers. The frustration in Brussels during that time was: ‘We don’t really know what you’re asking us for’. We only got Chequers in June and here we are.

“And to be honest, all the stuff about extended transition, all the stuff about backstop, that’s insurance policies because the future arrangements, the future relationship, is not nailed down because that bit hasn’t been really negotiated over the past two years. That’s a massive failure.”

If the deal was voted down, Starmer said it would probably take political pressure from MPs to avoid the alternative of a no deal, and the agreement of the EU to extend article 50.

“If you had a vast majority saying we don’t authorise the government to leave with no deal, it would be very difficult politically to do so. I accept that might not be legally binding,” he said.

“I accept that stopping no deal is something that’s going to have to involve the whole of the EU, but I think there would be a very strong push by the majority in parliament against no deal. And I don’t think this prime minister would simply plough on regardless, because she knows how dangerous no deal is.”

Quick Guide Why extend the Brexit transition period? Show Will the proposal solve anything? The mooted extension to the transition period is a new idea being put forward by the EU to help Theresa May square the circle created by the written agreement last December and the draft withdrawal agreement in March.

That committed the UK and the EU to ensuring there was no divergence between Northern Ireland and the Republic of Ireland.

But it also, after an intervention by the Democratic Unionist party, committed the UK (not the EU) not to have any trading differences between Northern Ireland and Great Britain.

The problem is that these are two irreconcilable agreements. They also impinge on the legally binding Good Friday agreement, which brought peace to Northern Ireland and in some senses pooled sovereignty of Northern Ireland giving people a birthright to be Irish or British or both.

If the UK leaves the EU along with the customs union and the single market then the border in Ireland becomes the only land border between the UK and the EU forcing customs, tax and regulatory controls. The backstop is one of three options agreed by the EU and the UK in December and would only come into play if option A (overall agreement) or option B (a tailor-made solution) cannot be agreed by the end of transition. The Irish have likened it to an insurance policy. The new EU idea is to extend the transition period to allow time to get to option A or B.

But an extension is problematic for Brexiters and leave voters, who want the UK to get out of the EU as soon as possible.

The Irish and the EU will also still need the backstop in the withdrawal agreement, which must be signed before the business of the trade deal can get under way. Otherwise it is a no-deal Brexit.

Extending the transition into 2021 would mean another year of paying into the EU budget. Britain would have to negotiate this but it has been estimated at anywhere between £10bn and £17bn. Staying in the EU for another year would also mean continued freedom of movement and being under the European court of justice, which Brexiters would oppose.

Some reports have said May plans to challenge Jeremy Corbyn to a TV debate on the merits of her deal, something the Labour leader has agreed to do. Barclay told Sky he could not confirm it.

Asked why May planned to send herself and ministers around the UK to sell the deal, when it was MPs who would decide, Barclay said: “MPs themselves gave this decision to the public, MPs voted to have a referendum, to give the public a say. So it’s quite right that we take our case to the public. It was their vote.”

Speaking later on Today, Barclay dismissed fears the UK could be endlessly stuck in backstop arrangements over issues such as fishing, which has been raised as a concern by the French president, Emmanuel Macron.

“It’s not in the EU’s interests for their own trade negotiations, because one of the key points when you’re negotiating trade is to be clear on the size of the market, and you couldn’t do that if it was unclear if the UK was in or out.”

Barclay said he hoped to be able to sell the deal to fellow Brexit-supporting Conservative MPs: “It’s a deal for Brexiteers like me which delivers on key things we campaigned for. It allows a skills-based immigration system, it allows an independent coastal state, so we have control of our fishing, we control of our money, border and laws.”