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.
The GCHQ claims 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 little evidence to back it up.
And British intelligence is clearly determined to avoid being drawn into what is an increasingly toxic row in Washington.
A Senate committee has concluded that there are "no indications" Trump Tower was under surveillance by the US government before or after the election.
The statement from 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 maintained his claims.

The 49-year-old, a former international player, will take up his role in early January on a four-year contract.
Wong is currently women's singles coach to the Badminton Association of Malaysia (BAM).
He has previously worked with world number one Lee Chong Wei and his wife, Wong Mew Choo, the two-time Commonwealth Games silver medallist.
He has also worked for the last three years with world junior champion Goh Jin Wei, and bringing through Scotland's brightest youngsters will be a big part of his remit.
Badminton Scotland's chief executive Anne Smillie said: "It has been a long process but it was crucial that we took our time and found the right person, someone with the credentials and capability to develop our youngsters into fully-fledged internationals.
"In Tat Meng we believe we have found the person to do that. We have been fortunate in recent years to produce Olympic squad members and World, European and Commonwealth medallists and we want to continue that success story over the next four years and beyond."
Britain's communications intelligence agency GCHQ has issued a statement denying it wiretapped Donald Trump during the US presidential campaign.
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Badminton Scotland has appointed Malaysian Tat Meng Wong as the organisation's new head coach.