Sir Peter Westmacott, who was the UK’s top diplomat in the US until January, also urged the White House to pubicly accept British assurances that the allegation about GCHQ is nonsense.
White House press secretary Sean Spicer sparked anger among British intelligence chiefs last week by repeating an unsubstantiated claim on Fox News that British intelligence had spied on the president.
An allegation had also been made in a report on Russian TV channel RT that GCHQ passed intelligence on Mr Trump’s campaign to US intelligence bosses.
Asked about this on BBC radio, Sir Peter said: “We do know that the Russians are engaged in information warfare against the United States, the United Kingdom, a lot of western democracies.
“They have peddled stories in the past which have turned out to be not true and which were deliberately put out by them.
“So if the story comes from RT, it would indeed be part of the fairly standard Russian playbook with which all western democracies at the moment are having to cope.”
Sir Peter also warned that the claims against GCHQ, which the spy agency took the rare step to publicly deny, risked damaging intelligence sharing between the UK and US.
Meanwhile, the directors of the Federal Bureau of Investigation and National Security Agency were due to speak publicly today about the claims of alleged links between Russia and president Trump’s campaign.
FBI boss James Comey and NSA director Admiral Mike Rogers were due to be questioned during a rare open congressional intelligence committee hearing.
The Kremlin has denied that it meddled with the US elections.