Brexit talks have been deadlocked over the so-called backstop - the fallback position that would be activated if a future EU-UK trade deal does not produce a solution that would prevent a hard border between Ireland and Northern Ireland.
The agreement would avoid the need for an Irish backstop - which has left Britain and Brussels deadlocked - and would stop Northern Ireland being treated differently to the rest of the UK.
British Prime Minister Theresa May will bring plans to avoid a hard Border to her divided cabinet tomorrow.
Lisa Chambers, the Brexit spokesperson of Irish party Fianna Fáil, which similarly supports Varadkar's government in parliament, said that his willingness to consider a review process was a "significant and potentially hazardous change in direction".
Here, Labour leader Brendan Howlin said he remained "very fearful" Irish interests could be sacrificed by Europe to secure a deal and said Ireland has to be "vigilant".
The Sunday Times report said preparations for a final Brexit deal were "far more advanced than previously disclosed" and that May's agreement would satisfy both remain-voting Tories and the hardline Eurosceptics within her party.
A senior EU source said the United Kingdom prime minister would need determine whether she could sell an open-ended customs union to her party, but the calculation in Brussels was that it would be acceptable to the House of Commons as a whole.
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The official added that verbal orders went out this past weekend to some units and additional written orders are starting Monday. Fought back hard and viciously against Mexico at Northern Border before breaking through.
The paper also reported that the Prime Minister was on course to agree a future economic partnership that would leave open the possibility of Canada-style free trade deal sought by Brexiteers.
There appears to be a softening on the EU side to an all-UK customs union replacing the Northern Ireland specific text, but that would require an acceptance by Downing Street that it would in effect be a permanent arrangement.
This comes after backbenchers are pressuring May to ensure there is a time limit on the backstop. Yet it has been traduced and cast aside since the UK's Brexit referendum as a result of British indifference, unionist triumphalism and Irish powerlessness.
'The Irish stance is still the same and clear to say that an emergency time-bound plan or one that could be unilaterally completed by the United Kingdom will never be agreed upon by the country or the European Union, ' said Irish Foreign Minister Simon Coveney on Twitter.
In a letter to Mrs May they say Parliament should not be bound by the 2016 vote any more than it should be by the 1975 referendum that took Britain into the European Union, especially when there are question marks over its validity.
Even if a deal is done in Brussels in the coming days, May will have to sell it in London - first to her own Cabinet, and then to Parliament.