Two of the three Lloyds TSB branches in SE1 are part of the package of 632 branches to be rebranded as TSB and transferred to the Co-operative Group in a deal announced this week.
Lloyds Banking Group, which was rescued by the Government at the height of the financial crisis, must sell part of its business to meet European rules on state aid and has now agreed non-binding heads of terms with Co-operative Group plc.
The banks to be sold include the South Bank branch at the Shell Centre in York Road and the Elephant & Castle Shopping Centre branch. From next summer they will be rebranded as TSB before the transfer to the Co-op is completed in November 2013.
Of Lloyds TSB's three SE1 banking counters, only the London Bridge branch in Borough High Street is unaffected by the announcement.
Both the South Bank and Elephant & Castle branches are situated in premises due for redevelopment within the next few years.
"Today's agreement is an important step in meeting our obligations under the mandated sale of our branches," said Lloyds Banking Group chief executive António Horta-Osório.
"We believe the Co-operative will be a good owner for our business, customers and colleagues, and the combined banking business will be a significant competitor on the high street with nearly 10 per cent of today's UK branch network.
"In agreeing to move ahead with the Co-operative we provide greater certainty for our customers and for our shareholders. In addition to an upfront consideration, we will also get to share in the future financial performance of the combined banking business which will be an effective challenger with a strong customer focus."
Peter Marks, chief executive of the Co-operative Group, described the proposed deal as "the biggest shake-up in high street banking in a generation".
He added that the acquisition of the extra branches would "make the services of our member-owned, customer-led, ethically-driven, bank available to millions of people we've not been able to serve up until now".