It also follows a series of warnings that a deal could put the Northern Irish peace process at risk by undermining the United Kingdom government's neutrality.
Varadkar, at a news conference with May in Downing Street, said both the British and Irish governments needed to be impartial actors in relation to Northern Ireland's power-sharing arrangements, which are now stalled.
Speaking in Downing Street alongside Mr Varadka, Mrs May dismissed these concerns as she claimed her government "remains absolutely steadfast" in its commitment to upholding the Good Friday Agreement.
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".
The comments were seen as a coded reference to the party's opposition to scrapping the "triple lock" on pensions and means testing the winter fuel allowance - both of which were in the Conservative manifesto.
After the Tories' disastrous General Election showing saw the party lose its Commons majority, Mrs May sought a "confidence and supply" deal with the DUP to prop up her minority administration.
The DUP is also looking for a more generous deal from the Treasury over the planned devolution of corporation tax powers, as well as cuts in air passenger duty.
The Times branded her administration the "stumbling husk of a zombie government" and said May was now "so weak that she can not arbitrate between squabbling cabinet ministers", who are increasingly publicly divided over Brexit.
UK PM May forms minority govt after election gamble backfires
As one teaching student at the University of London told VOA, "I think most of us here were against Brexit past year ". Two weeks earlier, a suicide bomber killed 22 people as they were leaving an Ariana Grande concert in Manchester.
Both leaders expressed confidence that the Stormont institutions could be up and running again by the deadline of June 29, averting a return to direct rule from Westminster.
"We think it is very important that Northern Ireland should have a unique voice at this very important time", he said.
"What I think we need to do is to build a new deep and special partnership with the European Union", she said.
Northern Ireland has been without a powersharing Executive since March and without a first and deputy first minister since January.
Both leaders re-affirmed their commitment to ensure that Britain's withdrawal from the European Union did not lead to a return of a "hard border" between Northern Ireland and the Republic.
The talks became so strained in the past few days that the DUP negotiators in Belfast refused to pick up the phone to the Prime Minister's team for 36 hours, The Daily Telegraph can disclose.
Prime Minister Theresa May presents her programme to parliament on Wednesday with a focus on Brexit and terrorism, despite suffering an election fiasco that British media said has turned her government into a "zombie".
"We want to ensure that as much as possible, while there many be a political border between our two countries, there should not be an economic border, and that any border that does exist should be invisible". "We will work hard every day to gain the trust and confidence of the British people, making their priorities our priorities".
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