Irish Prime Minister Leo Varadkar arrived in Washington this week as part of an annual trip ahead of St Patrick's Day.
Mr Trump said: "I'm surprised at how badly it's all gone from the standpoint of a negotiation". I missed it last time but I would have loved to have been there.
Mr Varadkar said: "I really hope you will be able to accept my invitation to Ireland very soon and I can guarantee you a very warm welcome is waiting for you, especially in Co Sligo and Co Clare from where your forebears come".
One of Mr Trump's golf courses is in the County Clare village of Doonbeg.
Trump said that Brexit is "tearing countries apart".
"It's a very complex thing right now, it's tearing a country apart".
"People often ask me what I felt on Inauguration Day when I raised my right hand", Pence said.
The two leaders discussed Brexit as well as a number of Irish-US specific matters.
Government sources confirmed that Taoiseach Leo Varadkar has been told that congressman Richard Neale will reintroduce the E3 scheme and has said so to senate minority leader Chuck Schumer.
The Law Society of Ireland is visiting the USA to promote Ireland as an global legal centre among American lawyers and corporations, flagging its post-Brexit ambitions.
"We have a different opinion, president", Irish Prime Minister Leo Varadkar said mildly.
He added that Mr Trump was supportive of the peace process in Ireland and that he wanted to avoid a hard border.
"I regret that Brexit is happening and the United Kingdom was a really important part of the European Union".
"Both of us thought it was a good idea so we talked about how that might work", he said. "But I think it could have been negotiated in a different manner".
Varadkar even spoke about his sexuality at the Naval Observatory.
"We talked about immigration". He also said Pence and his wife told him that he and his partner were "very welcome to visit their home in the future".
The Taoiseach and Dr Matt Barrett met Mr Pence and his sister Anne at the vice-president's residence in Washington DC on Thursday morning.
Pence tweeted that he was honored to host Varadkar at his home.
However, it was not an occasion Mr Varadkar could allow pass without using the opportunity to highlight his own story and how he had, as a gay man, become the leader of a changed, tolerant Ireland. "I stand here leader of my country, flawed and human, but judged by my political actions and not my sexual orientation, my skin tone, gender, or religious beliefs".
That's a subtweet if I've ever seen one. It is found in every country where freedom and liberty are cherished.
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