Mr Kenny has previously stated his desire for the Irish government to play the strongest possible role in the Brexit negotiations and “make the strongest presentation for continued support for Northern Ireland”.
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Asked specifically whether he, like Leo Varadkar, would like to see a 32-county Republic in his lifetime, Mr Kenny hesitated, before replying: “My focus today is on confirming that there will not be a return to a hard border, by that I mean customs posts all along the way”.
Speaking in Newry, Mr Cox said work needs to begin on common standards, data and information sharing, e-border initiatives and reinforced co-operation between the UK Border Agency and the Irish Naturalisation and Immigration Service.
“It’s in all our interests to work together to safeguard out national security and the outcome of the referendum will not undermine it”, she said.
Confirming that Ireland would not be following Britain’s lead on Brexit, the Taoiseach said: “For our part, we have already made very clear our view Ireland is very much committed to staying in the EU”.
“Indeed, it’s vital that we keep up the momentum on tackling paramilitary groups and building a shared future”, Mrs May added.
“And today we have reaffirmed our commitment to establishing a new Independent Reporting Commission by the end of this year, which will support these efforts”. “We are in agreement that we don’t wish to see any return to the borders of the past on the island of Ireland”.
Both leaders have also met the North’s First Minister Arlene Foster and Deputy First Minister Martin McGuinness who were on separate sides of the European Union debate.
He is the first head of State to visit Downing Street since Ms May became prime minister.
The newly-appointed Mrs May also received “congratulatory” phone calls from the leaders of New Zealand, Japan, India, Pakistan and Jordan on Tuesday, a Number 10 spokeswoman said.
“It is testament to the importance of the relationship between the United Kingdom and Ireland that Enda is amongst the first leaders that I have met since I took office”, May said.
Mr Kenny insisted he has secured full agreement with the prime minister about the border issue as he moved to dampen concerns about the prospect of checkpoints.
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Mr May also offered Shinzo Abe her condolences following the knife attack at Sagamihara.
