Harding is set to step down in the spring after just over two years at Sainsbury's to head broadband provider Talk Talk.
Twigger would take the reins on Harding's departure, but would not become a board member as Harding was, claimed a source close to the retailer.
Harding had been approached by Talk Talk, he added. "She's very driven and ambitious. Justin King isn't going anywhere and his job is the only one she would have wanted."
Sainsbury's said it would confirm Harding's successor next month.
The retailer has opened 19 Locals in the first half of this year and is on track to open 50 c-stores by the end of the financial year and a further 100 in 2010/11.