The company wants to target existing and potential customers with products such as car insurance and foreign currency.
Joshua has managed the business for the past three years. It is not known which agencies are pitching, but it is likely that Joshua and Publicis Dialog are involved.
A Post Office spokesman confirmed the pitch was going ahead and that it had asked 10 agencies to respond.
The Post Office's expansion into the financial services sector is the result of a £125m deal with the Bank of Ireland.