There would be no point in sending Brexit negotiators back to Brussels or extending article 50 because no other withdrawal deal is on offer, Theresa May’s de-facto deputy has said.

David Lidington said critics in his own party and on the Labour benches who thought a better deal could be struck were wrong.

“There is no point in sending negotiators back, when negotiators on the other side are making clear that this is the deal on the table, this is the choice,” the cabinet office minister told BBC Radio 4’s Today programme. “I don’t really see that extending article 50 gets us anywhere. The EU has made its position very clear.”

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.

Lidington’s comments came as the former defence secretary Michael Fallon, once an ultra loyalist, said he could not back the deal unless it was renegotiated to guarantee frictionless trade and better access to new trade deals outside Europe.

Fallon, who resigned from the cabinet over harassment allegations last year, had been one of May’s most disciplined cabinet ministers, nicknamed “the minister for the Today programme” for his willingness to go on air to defend the government.

Fallon told Today on Tuesday the deal was doomed, and it would take another two or three years of negotiations before businesses had certainty about the future relationship.

He suggested article 50 had been triggered too hastily and that the two-year negotiation period should be extended.

“This is not a good deal and we need a better deal and if it is possible to send the negotiators back to Brussels for two or three months, or postpone the leaving date for two or three months, I think that would be in the long-term interests of the country,” he said.

“It was one of the mistakes from the beginning, opening negotiations before there was a completely agreed government position on what we do want.”

Fallon, who campaigned to remain in the EU, said he did not want a second referendum. “I think we should respect the decision of the British people and I think we should be leaving. We committed to that in our manifesto but we need to be very careful what we are leaving to go to.”

Lidington played down the implications of an intervention by Donald Trump, who said May’s agreement sounded like a “great deal for the EU” and that it could prevent the UK trading with the US.

“I don’t think that the prime minister meant that. And, hopefully, she’ll be able to do something about that,” the US president said on Monday.

Lidington said the comments were not “terribly unexpected” but that Trump was wrong to suggest a trade deal could not be negotiated, even if parts could not be implemented during the transition.

“It’s always going to be challenging to do a deal with the United States,” he said. “The US is a tough trade negotiator. President Trump has always said he puts America first. I expect a British prime minister to put British interests first but it will be a tough negotiation.”

Fallon said Trump should be taken at his word. “It’s no use us brushing it off and saying: ‘Oh no, we can do a deal with America’. He is the president of the United States and if he says it is going to be difficult then it certainly looks like it is going to be difficult.”

Under the negotiated deal, the UK will not be able to pursue an independent trade policy during the 21-month transition period after Brexit.

In the likely scenario that a EU-UK trade deal is not close to ratification by July 2020, the two sides will jointly decide whether to extend the transition period for up to two years. Alternatively, they could allow the “backstop” solution to come into force in January 2021.

Under the latter scenario the whole of the UK would stay in a customs union with Brussels to avoid a hard border in Northern Ireland. This would prevent the UK pursuing any trade deal covering goods – although agreements covering services would be allowed.