Britain has been warned by the Irish government that any attempt to fortify the border with the Republic to prevent migrants slipping into the UK by the back door "won't work".
Irish foreign affairs minister Charlie Flanagan insisted the two countries must keep the "invisible" border that exists at present after Brexit.
"It is absolutely essential that every effort be made to ensure the existence of what is an invisible border.
"So, any suggestion that there will be a heavily fortified EU frontier, or a heavily fortified border, be it for customs and trade on the one hand, or for security and immigration on the other, is simply inoperable. It won't work," Mr Flanagan told the BBC.
The foreign minister also expressed concern at reports International Trade Secretary Liam Fox is pressurising Prime Minister Theresa May to pull a post-Brexit Britain out of the EU customs union so the country can cut better global deals.
"I have to say I was very surprised at the comments attributed to Dr Liam Fox, and it would be a matter of concern to Ireland were the UK to withdraw entirely from the customs agreement.
"I believe it would result in a situation where there would be a lot of paper work, and consequent red tape. We need to minimise incumbents and bureaucracy," he said.
Former northern Ireland secretary Theresa Villiers, who was a prominent Leave campaigner, also said the border should remain open, despite the risk of illegal immigration.
"The reality is that would mean there would be, perhaps, some risk that non-Irish EU citizens might enter the UK over that land border.
"But the way you tackle people who come and work in the UK without the appropriate permissions is through measures such as cracking down on employing illegal workers.
"The border between the UK and Ireland has never been policed in a significantly hard way, even during the height of the Troubles, it was very much a free flowing border. The common travel area survived the Troubles.
"So, I think the idea of imposing, suddenly, a whole host of new border checks, frankly isn't practical and it's certainly not desirable."