The unusual move by the agency came after White House Press Secretary Sean Spicer cited claims first made on US TV channel Fox News earlier this week.
GCHQ responded by saying the allegations were "nonsense, utterly ridiculous and should be ignored".
Mr Trump has claimed that Trump Tower in New York was under surveillance.
But he has provided no evidence for the claim.
A Senate committee on Thursday concluded that there were "no indications" Trump Tower was under surveillance by the US government before or after the election.
'Chain of command'
The claims of GCHQ involvement were initially made by former judge Andrew Napolitano.
Mr Spicer quoted Mr Napolitano as saying: "Three intelligence sources have informed Fox News that President Obama went outside the chain of command.
"He didn't use the NSA, he didn't use the CIA, he didn't use the FBI and he didn't use the Department of Justice, he used GCHQ."
A GCHQ spokesman said: "Recent allegations made by media commentator Judge Andrew Napolitano about GCHQ being asked to conduct 'wiretapping' against the then president-elect are nonsense.
"They are utterly ridiculous and should be ignored."
It is unusual for GCHQ to comment directly on a report about its intelligence work, normally preferring to stick to the policy of neither confirming nor denying any activity.
But the allegations are so sensitive that the agency clearly felt they could not let them go unchallenged.
Donald Trump's claim that the Obama administration had ordered surveillance on him has generated enormous attention but with so far no evidence to back it up.
And British intelligence is clearly determined to avoid being drawn into what is an increasingly toxic row in Washington.
On Thursday Senate Intelligence Committee chairman Richard Burr dismissed Mr Trump's claim his phones were tapped.
Mr Trump had accused his predecessor Barack Obama of wiretapping Trump Tower during the presidential race.
Mr Spicer said Mr Trump stuck by his allegations.
Source : BBC
