Under questioning from Fianna Fáil deputy leader, Dara Calleary, Mr Coveney said the Government would not tolerate a situation where border controls would be moved to mainland Europe to protect the single market.
Speaking alongside French president Emmanuel Macron in Paris, Varadkar said there was still sufficient time for May to present "credible" solutions.
"A long extension, implying the United Kingdom takes part in European elections and European institutions, has nothing easy or automatic about it", Macron said. "I am repeating it here very strongly".
"The European Union can not forever be hostage to the resolution of a political crisis in the United Kingdom", he said.
"If the United Kingdom is not capable, nearly three years after the referendum, of coming forward with a solution that is supported by a majority, it will have effectively chosen a no-deal exit on its own", Macron said.
Meanwhile, 34 per cent of Labour supporters and 27 per cent of Liberal Democrats say they UK's current set-up should remain.
May grapples with Brexit deadlock as European Union warns of no
The call for a unified approach followed a nine-hour Cabinet session as senior ministers hammered out a road map. Parliament on Monday rejected four alternatives, with proposals for a customs union failing by three votes .
Mr Macron said that credible justifications for an Article 50 extension for the United Kingdom could include an election, second referendum or alternative proposals for the future relationship, such as a customs union.
Struggling to find majority support for her Brexit deal, May said on Tuesday she would ask the European Union for a further delay to Brexit beyond April 12 in a fresh attempt to end the deadlock.
"We are working very closely with Irish authorities to try and perform controls away from the border if at all possible", an European Union official, who did not wish to be named, said last week.
However, he also said that "there is still time for (Theresa May) to come to the European Council with proposals, proposals that are credible and have a clear pathway to success".
German Chancellor Angela Merkel is due to meet Leo Varadkar in Dublin tomorrow to discuss no-deal preparations.
In London, prime minister Theresa May said yesterday she would ask the European Union for a further delay to Brexit beyond April 12 to give her time to sit down with the opposition Labour Party in a bid to break the impasse over Britain's departure.