The SFO and Tesco both declined to comment on the report by Bloomberg, which said that Mr Clarke and former commercial director Kevin Grace had been asked to come in for interviews under caution.
A source also told the news service that ex-UK finance director Carl Rogberg had been interviewed earlier this summer and former head of UK operations, Chris Bush, had also received an SFO request.
Bloomberg reported that all of the former executives either declined to comment directly or through their lawyers.
The SFO launched a probe last October after the discovery of a £263 million hole in profit expectations at Britain’s biggest supermarket. It was later found to be £63 million bigger than this.
Details of the scandal - which involved rebates from suppliers being moved around to different periods on the company’s balance sheet - emerged last autumn, just weeks into the tenure of new chief executive Dave Lewis.
Mr Lewis had taken over after sliding sales prompted the departure of Mr Clarke. Tesco faces potential legal action from institutional shareholders over the accounting scandal.
In February, the supermarket reversed its decision to suspend more than £2 million in severance payments due to Mr Clarke and former finance director Laurie McIlwee as it sought to avoid a costly legal action.