Trump, who met May last summer at Blenheim Palace, England, said he had given her his ideas on how to handle the negotiation on Britain's break from the European Union but that she had not taken his advice.
President Trump, perhaps not wanting to spoil the chummy mood of the annual St. Patrick's Day-themed visit, announced at the start of his meetings with the Irish prime minister Thursday that he wasn't going to comment on Brexit, on which he and his guest vigorously disagree.
Trump said Brexit was "tearing countries apart" and added that he was surprised "how badly" it has gone since the UK's referendum on leaving the EU. "But I think it could have been negotiated in a different manner, frankly".
Asked whether he thinks the United Kingdom should push back its exit date, Trump said: "I think they're probably going to have to do something, because right now they're in the midst of a very short period of time - the end of the month - and they're not going to be able to do that". Parliament voted later in the day to request a delay of the planned March 29 exit. Lawmakers have committed the country to staying in the bloc unless a divorce deal is ratified.
He said he would like to see the "whole situation with Brexit work out", adding "we are talking with them about trade and we can do a very big trade deal with the UK".
But he now he says: "The potential is unlimited!" "I predicted it was going to happen".
"And I don't believe my country is the only one in the world where this story is possible".
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The Taoiseach referred to Mr Trump's "Make America Great Again" slogan and said that while the U.S. had military might and a booming economy, they shouldn't lose sight of what makes America great already - its people and its values.
MPs have rejected May's withdrawal agreement by huge margins twice and will next vote on delaying Brexit on Thursday night.
Nile Gardiner, a former aide to British prime minister Margaret Thatcher and director of the Margaret Thatcher Center for Freedom at the Heritage Foundation, said Trump "has been very clear in his view that Brexit is great for America and for Britain". And also we want to make sure we still have frictionless trade between Britain and Ireland because I believe in free trade.
"The Irish government has played a very adversarial role", Gardiner said. Pence got the pronunciation right - "TEE-shuck" - and said he, too, plans a trip to Ireland shortly.
Varadkar was returning to the White House for an evening reception and the presentation of a bowl of shamrocks to Trump.
Mr Trump has said he will be visiting the Republic of Ireland at some point this year. "Everything does", he said. "Everything does. One way or the other, it's going to work out".