Singapore's biggest lender DBS Group Holdings
Financial details of the transaction were not immediately clear, though one of the people said the deal could fetch about $200 million.
An Indian private sector lender is also likely to bid for the unit, another source directly involved in the process said, declining to give details. The sources declined to be named as the talks are not yet public.
RBS, 73 percent owned by the British government, said in February it would shrink its banking operations, pulling out of about 25 countries including India to help it refocus on lending in Britain.
The India business of RBS comprises corporate banking, trade finance and cash management.
DBS, FirstRand and RBS declined to comment.
($1 = 0.6497 pounds)