It is understood that Biggs has not yet formally accepted the job but he will be tempted by the chance to lead a firm that is set to enter the FTSE 100 when it floats either later this year or in early 2013.
RBS declined to comment on the report.
Biggs is currently chairman of takeover specialist Resolution, created to buy underperforming UK life insurers and merge them into a more profitable whole, which has taken over parts of AXA and Bupa.
RBS was ordered by European regulators to offload its insurance division as payback for state aid it received in the 2008 crisis.
The bank has until the end of 2013 to sell or float the business, which includes Britain’s biggest motor insurer.
The unit was renamed Direct Line Group last month after its best-known Direct Line brand but also owns home and motor insurer Privilege and the Green Flag roadside recovery service.