Update 9.06pm: Mike Pence has told Leo Varadkar his partner, Matthew Barrett, would be welcome in his home next year.
Asked about his meeting with Vice President Mike Pence, Mr Varadkar said: "We had a really, really good breakfast this morning with Mike Pence, his wife and his family, got to meet his mother.
"His mother really knows Ireland very well. Her father was born in Ireland."
The Taoiseach said she is even able to tell the Humpty Dumpty nursery rhyme in the Irish language.
He described it as a really nice engagement with the Pence family.
"I did privately manage to speak to them about equality and my support for equal rights for women and the LGBT community here in America and also in Ireland," Mr Varadkar said.
"They were very well-briefed. They knew about my personal story, they knew that my partner was living in Chicago and they said that both Matt and I would be welcome to visit their home in future, so I thought that was a very nice gesture.
"There are so many ifs and maybes, first of all I have to survive another year in my current office and, secondly, Matt is not terribly keen to attend official functions but you never know."
Update 6.09pm: Mike Pence has told Leo Varadkar his partner would be welcome in his home next year.
Leo Varadkar discussed LGBT rights with the Vice President during a meeting at the White House yesterday.
At a breakfast in the Vice President's residence this morning, Pence told the Taoiseach his partner Matt would be welcome to come with him next year.
Mr Pence has been an opponent of laws promoting marriage equality in the past.
Original story (4.04pm): Taoiseach understood to have raised LGBT issues with US VP Pence
By Elaine Loughlin, Political Correspondent
The Taoiseach has had a “tete a tete” with the US Vice President Mike Pence.
Leo Varadkar is understood to have raised LGBT issues with the Vice President.
Mr Varadkar met Mr Pence, who is known for his conservative views on marriage equality and other LGBT issues, at a breakfast in Washington this morning.
The annual breakfast meeting in the Vice President’s residence, which is part of the Taoiseach’s St Patrick’s itinerary, was not open to the press this year as has been the case in previous years.
Speaking at the breakfast Mr Varadkar referred to Mr Pence’s Irish Heritage .
He said: "I understand that as a child Vice-President Pence could recite the nursery rhyme, ‘Humpty Dumpty’ in Irish after learning it from his grandfather. I won’t embarrass him by asking him how much he remembers. I won’t embarrass myself by admitting I only know it in English.
"As you know, the Vice-President’s grandfather, Richard Michael Cawley, was a remarkable man who left Doocastle in Co. Mayo, near Tubbercurry in Co. Sligo, as a young man because his mother had faith he would find a future here. A patriot and a proud family man, he built a new life in Chicago, and raised his children to love both Ireland and the United States. His mother’s dream was fulfilled.
"Mr Vice-President, you have spoken also of your own visits to Ireland, both as a young man and then later, bringing Karen and your children to Sligo.
“Not to forget, of course, your visits to your cousin’s bar in Doonbeg, now in direct competition with your boss’s slightly larger establishment across the street,” he said in front of invited guests.
On trade and the countries' wider relationship, Mer Varadkar added: "Here in the US, over 100,000 Americans work in Irish-owned companies, employing people in every one of the 50 states. Our two countries trade nearly $2bn worth of goods and services every week of the year.
"It is clearly in our mutual interests not only to sustain this trade and investment, but to continue to grow it in the years to come.
"One of the most meaningful Irish proverbs is ‘Ní neart go chur le chéile, ‘together we are strong’. This is as true today as at any other point in our history."
During the speech, the Taoiseach invited Mike Pence to visit Ireland.