In an incident which triggered a huge police response, raised alarm among allies and remains largely unexplained, five British men in their 20s were arrested on Sunday on suspected terrorism and explosive offences near RAF Fairford in western England.
The five, who all live in London, were later released on bail.
"There are strong indications that Iran played a part in what happened over the weekend at RAF Fairford but of course it remains an ongoing, complex and serious police investigation," Burnham told the BBC in an interview.
On Monday, Iranian authorities strongly denied any involvement, with the embassy responding to reports that it could be involved in the incident.
Burnham said the UK was working closely with the US, whose bombers have in recent months used the base to launch missions against Iran.
Shortly after the arrests, US President Donald Trump said the men had planned "big damage" and his Secretary of State Marco Rubio later pointed at "a foreign actor".
Until Burnham's remarks, the UK had not linked the incident to Iran although counter-terrorism police have said their investigation included the possibility of foreign state involvement and whether the incident could involve proxies "knowingly or unknowingly working on behalf of a foreign state".
Police said on Tuesday that petrol was found in the vans used by the men but no improvised explosive device.
Police also defended their decision to release the men on bail.
Adding to the intrigue around Sunday's events, The Times reported that one of the men called emergency services shortly before being arrested in the early hours.
"Whether it was surveillance, a 'dry run' to test security or a stunt to send a message to the British government are all under consideration," The Times said, adding that investigators were unsure whether he got cold feet or it was part of a plan.
The newspaper did not cite its sources.
Police declined to comment on whether one of the suspects had made the call.
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