Reporters in the Oval Office said Mr Varadkar kept the U.S. president waiting on the phone for around 90 seconds before answering.
Both leaders said they look forward to meeting in person.
An Irish government spokesperson confirmed the planned call, adding that Varadkar had held phone calls with a number of other global leaders since his appointment two weeks ago - including French President Emmanuel Macron, British Prime Minister Theresa May, Indian Prime Minister Narendra Modi and German Chancellor Angela Merkel.
Earlier, Mr Trump will have a conversation with French President Emmanuel Macron, receive his daily intelligence briefing and meet with his National Security adviser HR McMaster.
During the phone call today, President Trump requested a broader explanation on how the border works between Northern Ireland and the Republic of Ireland - something which he also asked of former Taoiseach Enda Kenny when he met him last St Patrick's Day.
Victory in an electoral college system is something Trump understands. I know so many of them, I feel I know all of them.
The 38-year-old Varadkar is Ireland's youngest prime minister.
"But I just wanted to congratulate you; that was a great victory that you had".
"I will not, of course, rescind that invitation", he said, adding that to do so would be "inappropriate" and would create a "diplomatic incident". However, he said no progress had been made in organising the visit and there was no timeline in place.
Mr Varadkar also said he would "never shirk" from raising issues such as climate change with the US Government and its representatives in Ireland.