The Irish prime minister, Leo Varadkar, has said the revised Brexit deal does not undermine the backstop nor reopen the withdrawal agreement.

The backstop – an insurance policy to avoid a hard border on the island of Ireland – will continue to apply “unless and until” it is replaced by future arrangements that ensure no hard border, he said in a statement on Tuesday morning.

Addressing the media in Dublin, the taoiseach welcomed the agreement reached between the UK and the EU on Monday night as “positive” and urged the House of Commons to vote for it on Tuesday night to lift the “dark cloud” of Brexit.

He echoed Jean-Claude Juncker, the European commission president, in saying that a freshly negotiated legal add-on to the Brexit deal was “complementary” and not a rewrite.

“It does not reopen the withdrawal agreement, or undermine the backstop or its application. It says that we will work together, in good faith, in pursuit of a future relationship that ensures that the objectives of the protocol, particularly the need to avoid a hard border, are met.”

Varadkar read from a prepared statement and did not take questions, reflecting the Irish government’s desire to project a sober tone and not say anything to complicate Theresa May’s uphill battle to get the deal through Westminster.

The taoiseach did not call the backstop “cast iron” and “bulletproof”, as he did in December 2017, comments that played well in Ireland but angered Brexiters.

In an effort to reassure Brexiters, including the Democratic Unionist party(DUP), he committed the Irish government to exploring alternatives to the backstop “in a timely way” if talks over the future relationship between the UK and EU break down.

“The further texts agreed yesterday provide the additional clarity, reassurance and guarantees sought by some to eliminate doubt or fears, however unreal, that the goal of some was to trap the UK indefinitely in the backstop. It is not. Those doubts and fears can now be put to bed.”

Varadkar was at Dublin airport preparing to fly to Washington on Monday night but returned to government buildings for a late-night cabinet meeting.

Behind the Irish government’s outward calm one official said there was deep anxiety and nervousness that Westminster will reject the deal, putting the UK on course for possibly leaving the EU without a deal, and that Ireland’s insistence on the backstop will be blamed.

Lisa Chambers, Brexit spokesperson for the main opposition party, Fianna Fáil, said the EU had granted some leeway to May. “It’ll be that little bit easier for the UK to exit the backstop,” she said.

Jeffrey Donaldson, a DUP MP, offered some hope for the deal when he told RTE his party was examining the documents and was awaiting advice from the attorney general, Sir Geoffrey Cox.

Varadkar alluded to the title of Kazuo Ishiguro’s novel in saying the Irish government would try to refrain from further comment on Tuesday. “For the remains of the day, we need to give MPs in Westminster the time and space to consider what’s now on the table.”