Trump's press secretary, Sean Spicer, made the claims at a press conference last night, apparently referring to a report on Fox News.
Spicer told reporters that British agents had spied on Trump Tower during the election campaign.
GCHQ issued a strong denial this morning, with a spokesperson commenting: "Recent allegations made by media commentator Judge Andrew Napolitano about GCHQ being asked to conduct 'wire tapping' against the then President Elect are nonsense.
“They are utterly ridiculous and should be ignored."
Spicer and US national security adviser General McMaster have now apologised for the claims, according to a report in the Telegraph.
The allegations come weeks after President Trump accused his predecessor of tapping his phones in a series of tweets, although he gave no evidence for his assertions.
Terrible! Just found out that Obama had my "wires tapped" in Trump Tower just before the victory. Nothing found. This is McCarthyism!— Donald J. Trump (@realDonaldTrump) March 4, 2017
Former US national intelligence director James Clapper, who stood down at the end of the Obama administration, slammed Trump's claims.
"There was no such wiretap activity mounted against the President-elect at the time, or as a candidate or against his campaign," he said.