Theresa May has pledged that the details of any deal for the Democratic Unionist Party to prop her up in power will be made public, as talks go down to the wire.

No agreement has been finalised to guarantee the Conservatives a Commons majority – nine days after No 10 was forced, embarrassingly, to retract a claim that it had already been reached.

The continuing delay raises the prospect that the Queen’s Speech will go ahead on Wednesday without a deal with the DUP in place.

It is believed to have been caused by the DUP’s big financial demands, for both higher public spending in Northern Ireland and lucrative tax breaks.

At the weekend, John McDonnell, Labour’s shadow Chancellor, wrote to the Treasury over reports that the DUP is seeking an exemption from air passenger duty levied at airports in Northern Ireland.

If granted, that demand would cost at least £90m a year, Mr McDonnell said - enough to pay for “2,000 firefighters”.

Now the Prime Minister has bowed to the pressure to be fully open about the agreement – while apparently conceding it may yet not happen.

Following talks in Downing Street with new Irish Prime Minister Leo Varadkar, Ms May insisted any deal with the DUP would not be allowed to undermine the Good Friday Agreement.

And she said: “We continue our discussions with the DUP. We are talking about a confidence and supply agreement with them.

“On reaching such an agreement, we will make sure that the details of that are made public so that people can see exactly what that is based on.”

Some Conservative MPs have questioned whether the deal is even necessary, given the DUP’s determination that Jeremy Corbyn must be prevented from reaching No 10.

The Queen’s Speech would pass, albeit narrowly, even if the DUP merely abstained – rather than voted with the Tories – they argue.

The talks to achieve a “confidence and supply” arrangement with the right-wing Northern Ireland party have proved tortuous, despite Ms May’s hopes of a quick success.

It would mean the DUP backing the Government on its Budget and other key votes and – crucially – preventing it being brought down by motions of no confidence.

Nine days ago, Downing Street claimed an outline agreement had been reached, but was forced to backtrack after a DUP spokesman denied it.

Last week, Sinn Fein and the SDLP, as well as the cross-community Alliance Party, warned that a deal would undermine attempts to restore the power-sharing executive at Stormont.

Speaking alongside Ms May, Mr Varadkar said he was reassured by the Prime Minister's commitment to make public the terms of any agreement.

“We spoke about the very important need for both governments to be impartial actors when it comes to Northern Ireland and that we are co-guarantors of the Good Friday Agreement,” he said.

On the danger of undermining that peace process, he added: “I am very reassured by what the Prime Minister said to me today that that won't be the case.”