He noted that it was a “matter of public record” that activity by Russian media organisations, and “trolls and bots” had been detected during the referendum campaign.
However, he said he was “anxious that we don’t play into our adversaries’ hands by making these trolls look 10ft tall”, adding: “That is not our experience.”
Sir Ken added: “They’ll be delighted because they are front page news once again.”
In the House of Commons on Monday, Kemi Badenoch had accused the Prime Minister of having “implied that the referendum result was illegitimate because of Russian interference”.
The Government’s official position has always been that Russia did attempt to interfere in the referendum, but had not been successful.
Mr Burnham is pushing for closer ties with the European Union and will set out options for doing so after a summit with the bloc on Nov 20. His comments on Russia come two weeks after he suggested it was “possible” that Labour’s next election manifesto could include a pledge to hold an in-out referendum on the EU, or to rejoin the bloc without another vote.
Responding to Mrs Badenoch’s comments, Mr Burnham said he had implied “no such thing” and had “always respected” the outcome of the Brexit vote.
‘Russian actors sought to interfere’
He added that “it is a matter of record that Russian actors sought to interfere” in a previous general election and continued: “We think it is inconceivable that this was the first time.”
Sir Ken said he agreed “strongly with what the Prime Minister said yesterday” and added that there had been a “hack and leak” during the 2019 general election campaign.
On Tuesday, Luke Pollard, a defence minister, said the Government would aim to be more transparent by revealing how Russia had been behind a campaign to “misinform and divide” the public.
He added that the Government believed Russia had attempted to interfere with numerous general elections but had so far been unsuccessful. However, he warned that Russia’s efforts to intervene and interfere with Britain’s democracy were increasing.
Asked whether the Government would show evidence of this interference, Mr Pollard said it should follow the example of the Ministry of Defence, which he said was being more transparent with the public about the threat from Russia.