Shortly after the referendum, Republic of Ireland warned that setting old-fashioned hard land border would hinder the Northern Irish peace process.
Theresa May's allies at Westminster said they would mobilise their ten MPs to oppose the move, adding that the prime minister had "already reiterated this".
The Democratic Unionist Party (DUP) is said to be furious following the report, published Friday in the Times, that Ireland wants the Irish Sea to act as the post-Brexit border with the UK.
In the latest wrinkle, The Times newspaper reported that the Irish government's preferred option is for customs and immigration checks to be located away from the land border and at ports and airports.
Ian Paisley Jr, one of the 10 DUP members of the British parliament allowing May's government to stay in power, went on Twitter to condemn the idea.
But Dublin is unconvinced by the UK's plans to use technology to maintain the invisible land border between the north and south of the island. "At Westminster we will continue to use the influence of our 10 MPs to ensure that respect for the integrity of the United Kingdom remains at the core of the negotiations process", Dodds said in a statement published on the DUP website.
The Taoiseach revealed he and Foreign Affairs Minister Simon Coveney do not want to see the nation split in two when it comes to freedom of trade and customs issues.
The border between Republic of Ireland and Northern Ireland is one of the key issues requiring a solution before Brexit, since the United Kingdom will face nearly 310 miles long land border with the European Union.
Taoiseach Leo Varadkar really hammered home his firm opinion about not wanting any kind of economic border with the United Kingdom once they have finally carried out their Brexit plans.
DUP chief whip Sir Jeffrey Donaldson told the BBC'S Today programme that the idea of a sea border was "absurd and unconstitutional".
"That is not going to work".
"Therefore, we can not create a border between Northern Ireland and Great Britain", said the spokesman.
"There is no realistic solution that has been put forward", he said.
"This apparent hardening of attitudes within the Irish Government is untimely and unhelpful".
A Whitehall source said: "There is a new taoiseach and a new foreign minister and they're stamping their authority". The issue of how the Republic and Northern Ireland will fare after Britain leaves the European Union is particularly sensitive given the decades of violence in the province over whether it should be part of Britain or Ireland.
It comes after Home Secretary Amber Rudd pledged there would be "no cliff-edge" on freedom of movement after Brexit, as she outlined an "implementation" period where European Union nationals could register to come and work in the UK.