The EU would be "open" to such proposals, but it was for London to put them forward, he said.
However, the British government only has until the end of next week to come up with a plan - with European Union leaders to meet next Wednesday to discuss the situation.
"We don't want Ireland to be a back door to the single market, anymore than anybody wants us to become a back door to the single market", Varadkar said after a meeting in Paris with French President Emmanuel Macron.
"The European Union can not forever be hostage to the resolution of a political crisis in the United Kingdom", he said. "I am repeating it here very strongly".
"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", he said.
Meanwhile, 34 per cent of Labour supporters and 27 per cent of Liberal Democrats say they UK's current set-up should remain.
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But the British government has refused to envisage carrying out customs checks on goods arriving in northern Ireland from Britain, arguing that to do so would drive a wedge through the United Kingdom.
"However, there is still time for the Prime Minister to come to the council with proposals - proposals that are credible and have a clear pathway to success and we need to be open to any proposals she might bring forward to us".
Macron said the EU's priority was protecting its workings and the single market: "We have a future to build together in the European Union and a future relationship to build with the United Kingdom, which will be an ally, but we can't spend the next months still trying to fix the rules of our divorce and looking to the past".
The minister was referring to meetings that had begun at official level with the European Commission on how to manage the border in a no-deal Brexit, Reuters said.
Main image: Leo Varadkar and Emmanuel Macron in Paris today.
However Irish Foreign Minister Simon Coveney on Tuesday said Dublin will not countenance checks on its exports at European Union ports following any no-deal Brexit as a result of its plan to keep an open border with Northern Ireland.
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