At the British-Irish Council summit in the Isle of Man, Mr Lidington said: "The Prime Minister has always been very clear we won't accept something that involves carving out Northern Ireland from the rest of the United Kingdom".
"A successful outcome is not guaranteed but I think it is possible in the next couple of weeks", Mr Varadkar said.
The DUP, who are critical in helping the Tory minority government stay in power see this leaked letter as part of May's preparatory work, before a likely showdown with the party concerning checks in British ports or factories in Northern Ireland or Britain.
Visit: DUP leader Arlene Foster arrives in Dublin.
The Times newspaper reported that May sent a five-page letter on Tuesday to the leaders of Northern Ireland's small Democratic Unionist Party that props up her government.
But the DUP has interpreted the wording of her letter to mean that the measure will be contained in the Brexit divorce deal despite Mrs May's insistence it will never come into effect.
As we enter the endgame of the Brexit negotiation process, all pro-Remain parties must continue to represent the demand for our rights and political processes to be protected and the onus is firmly on the Dublin Government and the European Union 27 to ensure Theresa May is held accountable to the agreement she made in December.
It comes after the PM last month told MPs that 95% of the deal had been agreed, with the key sticking point of the "backstop" to prevent a hard border between Ireland and Northern Ireland remaining unresolved.
In response to the emergence of the letter, a Downing Street spokesman said: "The prime minister's letter sets out her commitment, which she has been absolutely clear about on any number of occasions, to never accepting any circumstances in which the United Kingdom is divided into two customs territories".
Wilson said: "She is now contemplating signing up to a legal agreement which, regardless to her aspirations, would be binding on the government of the UK".
The EU's proposed fall-back position to avoid a hard border - the so-called backstop - would effectively keep Northern Ireland aligned with Brussels's customs union and single market.
TAOISEACH Leo Varadkar has claimed that a deal between Britain and the European Union on Brexit could be possible within a fortnight.
This backstop - created to avoid a hard border on the island of Ireland - would be the last resort option and come into effect only if future trade negotiations failed to produce a solution for preserving the open Irish border.