He was due to arrive in November for a brief visit, either on his way to or from a World War One commemoration event in Paris.
Mr Trump was planning to spend one day in Dublin before travelling to Doonbeg, Co Clare where he owns a golf link.
The President is scheduled to travel to G20 meetings later in November in the Argentine capital, Buenos Aires, and make a stop in Colombia. "As details are confirmed we will let you know", White House spokeswoman Sarah Sanders told reporters.
"There is an open invitation to the US President to visit Ireland at any time", he added.
In March Irish Taoiseach Leo Varadkar and Trump met in the White House to discuss trade, immigration, climate change and human rights.
He had originally been invited by former taoiseach Enda Kelly on his final St. Patrick's Day visit to the White House previous year, an offer that was reiterated by Mr Varadkar last March.
A spokesman for the Irish Government said: "I can confirm that the proposed visit of the U.S. president is postponed".
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Trump announced in August that he would travel to Ireland as part of a trip to attend the November 11 commemoration in Paris of the 100th anniversary of the armistice that ended World War One.
And he pointed to the lack of a USA ambassador to Ireland for two years when he said: "I think it's time to try and get this relationship sorted".
It was to be Mr Trump's first visit to Ireland since he assumed the office.
He said the Government hadn't been alerted that the visit was on in the first instance.
Varadkar said then that Trump's decision to accept an open invitation "came a little bit out of the blue".
He said: "It will be controversial because everything Donald Trump does these days is controversial".