Theresa May’s plans to avoid a hard Irish border after Brexit are in fresh doubt after it was claimed she had asked the Irish government for help in finding a solution.

Leo Varadkar, the Irish Prime Minister, said the two leaders were now working together following a face-to-face meeting in Belfast at the start of the week.

The idea of some sort of “customs union partnership” comes after London rejected Dublin’s plea for the UK to remain in the EU single market and customs union, to avoid the need for a hard border.

Last week, Michel Barnier, the EU Brexit negotiator, warned that border checks would be “unavoidable” if the UK pursued its intention to leave the EU’s trading rules.

One Labour MP said the development showed the Government’s Brexit preparations reaching “peak farce”, adding: “They are asking other governments to come up with ideas instead.”

However, a government source downplayed Mr Varadkar’s comments, insisting No 10 did not recognise the suggestion he had been asked to help.

Mr Varadkar is coming under growing pressure in Dublin to show last December’s UK-EU agreement – that there will be no return to border posts and checks – will be adhered to.

In the Irish parliament, he insisted there would be little difference between the current customs union and any new partnership that was reached.

“We have been asked by the British authorities to work with them at official levels on how that might be achieved,” Mr Varadkar said.

It was “inevitable that border checks” would be introduced if the joint attempt to agree a customs partnership failed, he insisted.

Micheal Martin, the Fianna Fail leader, has raised fears that the border deal reached in December is not as “bulletproof” as it was described at the time.

It stated that, in the absence of an overarching EU-UK trade deal providing a solution, the entire UK would maintain “full alignment” with EU regulations.

But Ms May has downplayed the deal – once describing it as a “default default position” and refusing to say which goods would be covered by it.

Before Christmas, she also refused to rule out cameras at the Northern Ireland border, telling MPs: “We are not going to give a running commentary on every detail of our negotiations.”

Since then, the Prime Minister – under pressure from Brexiteers – has repeated her insistence that the UK will leave the single market and customs union regardless.

Stephen Doughty, a Labour MP and supporter of the pro-EU Open Britain group, said: “The Government’s total failure to deal with the realities of Brexit has now reached peak farce: they are asking other governments to come up with ideas instead.

“Leaving the customs union and the single market means creating a hard border on the island of Ireland. The idea that Theresa May is now begging the Republic of Ireland to solve a problem she has created is ridiculous.”

Tom Brake, the Liberal Democrat Brexit spokesman and supporter of the anti-Brexit Best for Britain campaign, said: “I am crossing my fingers that the Irish government can make the UK Government see sense.”

Mr Barnier has said “full alignment” is being drafted for the legally binding withdrawal agreement that the UK must sign in order to move to talks about the transition or future relationship.