Lloyds has ended its exclusive talks with The Co-op to sell 632 branches, allowing it to discuss a possible deal with rival suitor NBNK.
Lloyds said the exclusivity period with Co-op ended on Friday, and it will have detailed talks with NBNK if it is satisfied the start-up can get regulatory clearance for a deal.
That is likely to focus on the capital strength of NBNK and its ability to handle the job of running the IT systems needed for such a big branch network.
The sale, code named Project Verde, is likely to fetch 1 billion to £1.5bn and will create Britain's seventh-biggest bank. Lloyds has been ordered to sell the assets by European regulators as payback for being rescued by the British government during the 2008 credit crisis.
It has said it could spin off and float the business if it fails to find a suitable buyer.
The FSA discusses key issues with potential buyers and issues a "no objection in principle" guidance once satisfied.
Co-op's attempt to buy the branches has struggled to get past the regulator, due to concerns about its capital position and management.