The plan could be agreed at a European Council Meeting on 28 and 29 June in Brussels, the BBC has reported.
However, the British government has already voiced opposition to the idea of a single regulator and could opt out.
Meanwhile, it is also expected that a new bank bailout fund financed by a tax on financial institutions will also be announced, creating an EU-wide deposit guarantee scheme to protect savers in the event of a bank collapse.
A commission spokesman told the BBC that talks on the issues were “in the early stages”, although some elements of the banking union could be introduced without changes to existing treaties.