Dublin’s preferred option is for customs and immigration checks to be located away from the land border and at ports and airports instead, according to The Times.
But such a move would be unacceptable to the DUP, which the Prime Minister relies on to prop up her minority administration in the House of Commons.
DUP chief whip Sir Jeffrey Donaldson said: “There is no way that the DUP would go for an option that creates a border between one part of the United Kingdom and the other.”
He told BBC Radio 4’s Today programme: “Dublin really needs to understand that that proposition is absurd and unconstitutional.”
DUP MP Ian Paisley said Dublin’s position appeared to leave two alternatives - a “very hard border” or that “Ireland will wise up and leave the EU” itself.
The border between Northern Ireland and the Republic is one of the key issues that needs to be resolved by the UK and the EU before talks begin on a new trade deal.
British ministers had proposed using measures such as surveillance cameras to allow free movement between the north and south of the island.
However, sources have told The Times that Mr Varadkar thinks these plans could jeopardise the peace process in Ireland and restrict movement between the two countries.
He is said to want customs and immigration checks moved away from the land border to ports and airports - effectively drawing a new border in the Irish Sea.
Brexit Secretary David Davis has acknowledged that “flexible and imaginative solutions” will be needed to resolve issues around Northern Ireland although he has previously told MPs when asked about an Irish Sea border that “I don’t see that would be the solution, to be honest”.
Speaking to the Press Association earlier this week, Northern Ireland Secretary James Brokenshire insisted that “technology does have a role to play” in the future border arrangements.
He said it was important to view “the trading agreements as well as technology as a combined package together as to how we get that constructive agreement that does achieve what we want to achieve - which is that frictionless border between Northern Ireland and Ireland to ensure that we do not see the return to the hard borders of the past”.