Mandelson, who also advocates the part-privatisation of Royal Mail, suggested the 12,000-strong PO network could fill the gap left by the collapse of some high street banks.
Allowing post offices to offer mortgages, debit cards and other banking services would help to ensure the future of the network, he said.
Minister for Postal Affairs Pat McFadden said: "Lord Mandelson has said he wants to see a stronger role for the Post Office. I am warm about the idea. We have asked the Business and Enterprise Select Committee to look into this."
National Federation of Sub-Postmasters general secretary George Thomson added: "The Post Office is the natural home for increasing access to local banking and financial services. It has unrivalled geographical reach into communities which have long ceased to have a bank branch, as well as high levels of usage and trust."
Billy Hayes, general secretary of the Communication Workers' Union, said: "The people's bank is an idea whose time has come."