The opposition Fianna Fail party said on Thursday it would put a motion of no confidence in Deputy Prime Minister Frances Fitzgerald on Tuesday, a move that would breach the deal it agreed to support Varadkar's Fine Gael government in key votes for three years.
Fianna Fail indicated it might withdraw the motion if Ms Fitzgerald resigned, but Foreign Minister Simon Coveney told state broadcaster RTE that Fitzgerald would not resign.
The crisis comes weeks ahead of a European Union summit in which the Irish government has an effective veto on whether Britain's talks on leaving the bloc progress as it determines if EU concerns about the future of the Irish border have been met.
Brexit fallout
The EU has given Britain until early December to make sufficient progress on three key Brexit divorce issues - Northern Ireland, its exit bill and the rights of EU citizens living in Britain - in order to move on to trade talks at a summit on December 14.
Irish Foreign Minister Simon Coveney stressed that a solution to the Ireland problem was not only about protecting the Irish economy, but preserving a peace process that many in Dublin fear is being taken for granted.
"We are not in the business of building new barriers on the island of Ireland ever again," Mr Coveney said.
"What is at stake here, is not simply potential barriers of trade although that is a big concern, but it is also the potential destabilisation (of) a very delicate and quite fragile peace process," he said.
This view was echoed by his British counterpart Boris Johnson in UK parliament on Tuesday.
A return to a hard border, "would be unthinkable, it would be economic and political madness. Everybody understands the social, political and spiritual ramifications," he added.
But, in opposition to the Ireland and EU position, Johnson insisted that the solution to these problems would be properly addressed in the second phase of talks.
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