However, the Royal household declined to comment and so far the reports are unconfirmed.
Charles warned against "potentially catastrophic global warming" in a speech during a tour of the Caribbean in March.
Saying: "We demand the world's decision-makers take responsibility and solve this crisis.”
The Trump administration, by contrast, has been accused of deprioritizing the issue, withdrawing from the Paris climate accords in 2017.
It comes as Prince Charles became a grandfather for the fourth time after Prince Harry’s son, Archie, was born last week.
While the Irish government has played down speculation Mr Trump is embroiled in a spat over where he might meet Taoiseach Leo Varadkar, in the event he visits Ireland as part of his European tour.
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The anticipated stop would come at the end of his state visit to the UK and before he attends D-Day commemorations in France.
But with official confirmation still awaited from the White House, reports have emerged that officials are at odds on where Mr Trump might meet Mr Varadkar.
The president's visit to the island is set to be largely private, with Mr Trump expected to base himself at the golf resort he owns in Doonbeg, Co Clare.
Rumours of a disagreement focus on whether a meeting with Mr Varadkar would take place on Mr Trump's property at Doonbeg - the president's apparent preference - or on more neutral ground.
It has been reported that the Irish authorities would prefer the nearby Dromoland Castle.