At a meeting with the Business and Enterprise Select Committee set up to help secure the future of post offices, Post Office managing director Alan Cook said the move was "a return to old-fashioned banking".
"We have an aspiration to launch a current account next year, and I see no reason why that cannot be in every single branch," he said.
Cook added that the new Post Office Bank, which will include a debit/credit card, would not require a new banking licence as it would be run by an existing joint venture with Bank of Ireland.
National Federation of Sub-Postmasters chairman Geoff Thomson welcomed the plans, but called for the new bank to be fully owned by the government. He said: "This is the right thing to do, but we still need a government-backed bank at the Post Office."
The Communication Workers' Union (CWU) has demanded more government services be switched to the Post Office network. The CWU gave the Select Committee a list of services which it believes could be handled at post offices.
These included financial advice, passport interviews, help with filling in government forms, payment of court fines, tax self-assessment and repeat NHS prescriptions.
Speaking to the committee, head of the CWU Andy Furey said that post offices were "trusted and traditional outlets" for government services and the organisation's future was dependent on this role being developed.