Clarke and ex-commercial director Kevin Grace are being asked to come in for interviews under caution, Bloomberg says, citing people with knowledge of the probe.
The SFO opened a probe into Tesco last October.
The supermarket announced last September that it had overstated its profits by £263 million ($405 million). That figure has since risen to £326 million ($503 million), according to Bloomberg.
Clarke, who was a Tesco lifer who began stacking shelves at his local supermarket in the early 1980s, announced his resignation last July after a string of profit warnings. A whistleblower flagged the balance sheet misstatement months later.
Clarke's replacement Dave Lewis, who joined from Unilever, has since led a clearout of other executives who served under Clarke. The scandal resulted in Tesco's earlier this year posting the biggest ever loss by a UK retailer, £6.38 billion ($9.83 billion).
As well as the SFO probe, Tesco is facing a lawsuit from disgruntled shareholders. Accountancy watchdog the Financial Reporting Council (FRC) is also probing the role of Tesco's accountant at the time PwC in the scandal. And Britain's grocery watchdog is also probing the UK's largest supermarket over delayed payments to suppliers.
Tesco declined to comment.