Update 6.45pm: Donald Trump has claimed Enda Kenny is "a great guy", and says the Taoiseach is his new friend.
The US president made the comments after accepting an invitation from Enda Kenny to visit Ireland during his presidency.
The invite was offered by the Taoiseach during a half-hour meeting at the White House this afternoon.
Trump says he "loves Ireland", and that it is an honour to be the Taoiseach's friend.
President Trump said: "All of our friends, welcome. Taoiseach, that's my new friend, he's a great guy and you are something very special.
"We sat, we talked, and I think we're friends now too, right? It's really an honour."
Update 6pm: President Donald Trump has said the strong relationship between Ireland and the United States will grow closer than ever under his leadership.
Vowing to visit Ireland during his time in the White House, Mr Trump told Taoiseach Enda Kenny that he was his "new friend" and their respective governments would forge an even tighter bond in the years ahead.
Mr Kenny held a meeting with the president in the Oval Office during a packed day of St Patrick's engagements in Washington.
Afterwards they were hosted by Speaker Paul Ryan at his traditional St Patrick's lunch on Capitol Hill.
Speaking at the annual event, Mr Trump said he "loved Ireland and the people of Ireland".
"The people of Ireland and the people of the United States have stuck together through good times and bad times," he said.
"Over many centuries we have built a bond that thrives, inspires and endures and with us it's going to be closer than ever before.
"So as we celebrate our shared history and our enduring friendship let us commit ourselves to working together, as we will, to build on that bond to the benefit of our citizens for many more generations to come."
During the lunch, Mr Kenny made an impassioned plea to the president to help the 50,000 "undocumented" Irish who live in the United States without legal permission.
The Taoiseach has long campaigned for a legal pathway to be opened up to allow the undocumented to obtain legal residency without fear of deportation.
The issue has drawn intense focus on this year's visit, given the president's hard-line stance on immigration.
After the White House meeting, Mr Kenny said he and the president had agreed to work constructively on the issue.
Later, at the speaker's lunch, the Taoiseach stressed the important role played by so many Irish people through US history.
"This is what I said to your predecessor on a number of occasions - we would like this to be sorted," he told the president.
"It would remove a burden off so many people that they can stand out in the light and say 'now I am free to contribute to America, as I know I can'."
Mr Kenny also hailed those who continue to work for reconciliation in Northern Ireland, name-checking lunch guests Democratic Unionist MP Ian Paisley jnr and Sinn Féin President Gerry Adams.
Turning to the president, the Taoiseach said: "We want to protect this peace process and I know you are going to work with us in that context also."
There were lighter moments during the lunch, which was held before the Taoiseach's traditional presentation of a bowl of shamrock to the president back at the White House.
"They say the Irish have the capacity to change everything," joked Mr Kenny.
"I just saw the President of the United States read from his script ... entirely."
The earlier Oval Office meeting between Mr Kenny and Mr Trump was the new president's first face-to-face encounter with a leader of one of the 27 EU states that will remain in the union post-Brexit.
Mr Trump has been a vocal critic of the EU, having praised the UK's decision to leave, and the prospects of securing a bilateral transatlantic trade deal between Europe and the US appear to have receded under the new administration.
Mr Kenny said there was still the potential for negotiating new trade arrangements between the US and EU.
"Ireland will always be a friend of America, the European Union will always be a friend of America and that cooperation between these two most developed economies will be to the mutual benefit of millions of people in Europe and the United States," he said.
The meeting had the potential to be an awkward one, given that during the election campaign Mr Kenny accused Mr Trump of using "racist and dangerous" language.
Afterwards, the Taoiseach batted away a media question on whether the issue was raised during the bilateral discussions.
Ireland may be a small island, but look at all she has given us. Her light floods the world. pic.twitter.com/75Eenfeq9G— Paul Ryan (@SpeakerRyan) March 16, 2017
Mr Ryan spoke with pride of his own Irish roots as he addressed his lunch guests.
"Ireland may be a small island but look at all she has given us," he said.
Before toasting St Patrick by sipping from a pint of Irish stout, the speaker invoked the memory of the American revolution to pay tribute to Ireland.
"To America she is, as George Washington himself said, a friend of my country in my country's most friendless days - that says it right there."
Update 3.15pm: President Trump confirmed today he will make an official visit to Ireland, writes Fiachra O Cionnaith in Washington
He told the assembled media: “I love Ireland ..I’ll be there for sure.”
To which Taoiseach Mr Kenny swiftly added: “... during the course of his presidency.”
However, the media were asked to leave, with no further questions taken by the US President or the Taoiseach.
Update 1.45pm: Taoiseach Enda Kenny is holding a breakfast meeting with US vice-president Mike Pence at his DC residence this morning before his crunch meeting with US president Donald Trump this afternoon, writes Political Correspondent Fiachra Ó Cionnaith in Washington DC.
Mr Kenny and his wife Fionnuala arrived at just after 8.30am (12.30pm Irish time) before they were greeted by Mr Pence and his wife Karen, with Mrs Kenny joking that the "beautiful sunny day" is "very good for the shamrock".
During separate speeches just after 9.30am (1.30pm Irish time), Mr Pence said the visit of the Irish leader is a "very humbling experience" and revealed that in addition to two gifts he had been given the previous night Mr Kenny had also provided him with a framed 1911 census detailing information about his Irish grandfather who left in 1923 for the US.
Mr Kenny - who incorrectly referred to US president Donald Trump as "Mr Bush" - told Mr Pence that Europe and Ireland will "always be but a friend to the US".
He said he gave Mr Pence the gift of a hurley with his name in Irish on it, joking that Mr Pence said on receiving it that while the US army may have the best weapons he has been given the best tool to defend himself.
Earlier: Enda Kenny has held a breakfast meeting with US Vice President Mike Pence ahead of his first face-to-face encounter with Donald Trump.
The Taoiseach joined Mr Pence at his Washington residence for early-morning talks.
He will travel to the White House later today for a bilateral meeting with President Trump in the Oval Office.
Mr Kenny will then be guest at Speaker Paul Ryan's traditional St Patrick's lunch on Capitol Hill before returning to the White House for his annual presentation of a bowl of shamrock to the president.
The Taoiseach has pledged to raise with Mr Trump the case for legal recognition for the 50,000 Irish who live in the US without permission - the "undocumented".
Karen & I welcomed Taoiseach Kenny & Mrs. Kenny to our home for an Irish breakfast. Snow needs to melt before trying my new hurley. pic.twitter.com/Sr4uWT2yHL— Vice President Mike Pence (@VP) March 16, 2017
Mr Kenny is also the first leader of one of the 27 EU states that will remain in the union post-Brexit to meet Mr Trump.
The president has been a vocal critic of the EU, having praised the UK's decision to leave, and the prospects of securing a bilateral transatlantic trade deal between Europe and the US appear to have receded under the new administration.
The Taoiseach has said Europe needs to do more to convince the US leader of the worth of the EU project.
The encounter also has the potential to be an awkward one, given that, during the election campaign, Mr Kenny accused Mr Trump of using "racist and dangerous" language.
Yesterday, former Ukip leader Nigel Farage, a friend of Mr Trump, called on the Taoiseach to apologise to the president.
Mr Kenny rejected his remarks, insisting he was answerable to the Irish people, not Mr Farage.
The Fine Gael leader was keen to differentiate that his comments referred to Mr Trump's language, not his personality.
Last night, Mr Pence and Mr Kenny were guests of honour at a gala Irish- American dinner in Washington.
The Vice President delivered an emotional speech at the event, recalling his Co Sligo-born grandfather with fondness and claiming all his achievements in life were owed to his Irish heritage.
He also reaffirmed the "enduring commitment" of the United States to the peace process in Ireland.