Fox News commentator Andrew Napolitano alleged during a Fox & Friends broadcast earlier this week that Mr Obama had bypassed the US' intelligence community and used the UK's spy centre to obtain details of Mr Trump's conversations.
The claim was repeated on Thursday by White House press secretary Sean Spicer as he defended Mr Trump's still-unsubstantiated claim that Mr Obama "had my wires tapped".
It has prompted a forthright response from GCHQ. A spokesperson told CNN: "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."
The US Senate Intelligence Committee said earlier on Thursday there were "no indications" Trump Tower was under surveillance by the US government before or after the Presidential election.
President Donald Trump had used a series of Twitter posts to accuse his predecessor, Barack Obama, of wiretapping him.
In an interview on Wednesday with Fox News, Mr Trump suggested he first thought the former president was carrying out surveillance on Trump Tower after reading an article on the campaign in the New York Times.
Committee chairman Richard Burr, and vice chairman Mark Warner, said in a statement: "Based on the information available to us, we see no indications that Trump Tower was the subject of surveillance by any element of the United States government either before or after Election Day 2016."