The tradition, in which US presidents have been known to don green neckties, goes back to the 1950s and offers the small country special access to the US presidency at least for a day.
Around this time of year in 1952, then Irish ambassador John Hearne showed up at the White House to offer President Harry Truman some shamrocks, the symbol of Ireland.
It turned out Truman was on vacation, in Florida.
The next year Hearne did get to see President Dwight D. Eisenhower, who met with him for 15 minutes -- and accepted the bright, green leafy gift -- to talk about relations between the two countries.
The tradition has continued pretty much non-stop, helped by the fact that many US presidents have had some degree of Irish blood in them.
Ronald Reagan was particularly fond of the ritual.
"He was really big about St. Patrick's Day celebrations at the White House," said Matthew Costello, senior historian at the White House Historical Association.
"Reagan liked to tell Irish jokes about his family, his grandfather. He was really big about St. Patrick's Day celebrations at the White House,
"There's one year he snuck off and went down to Alexandria on St. Patrick's Day to have lunch and a pint with people. And they didn't tell anybody. They told the owners of the bar, like, an hour before that the president would show up," he added. The town is just across the Potomac River in Virginia.
Reagan and his sweet tooth - The visit by the Irish prime minister -- when Barack Obama was president, the water in a fountain on the White House lawn was dyed green -- is a chance to celebrate ties between two countries joined by waves of immigration from Irish people fleeing famine in the 19th century.
But it also provides a special diplomatic platform for a country of just five million people, an entree that small nations do not normally enjoy with the world superpower.
In the 1990s Irish leaders took this opportunity to press then president Bill Clinton to help nudge along the peace process in Northern Ireland.
Varadkar is expected to use his time with Trump to address diplomatic issues that include Britain's messy departure from the European Union and its impact on Ireland. The latter is part of the EU while Northern Ireland is in the United Kingdom. Their border is one of the main sticking points in the process.
On a personal level, the two men are very different. Trump is 72 and widely accused of being a white nationalist and anti-immigrant. Varadkar is 40, of partly Indian heritage.
Last year at this time they spoke about the situation of the many Irish living in America.
That came during the shamrock delivery session.
The plant is destroyed, for security reasons, said Costello.
But the president can keep the thing it came in.
Reagan used one such glass pot to hold the jelly beans he was so fond of, said Costello.
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