Lloyds Banking Group says it now has an "understanding" with the Co-op to sell it 632 Lloyds TSB and Cheltenham & Gloucester branches.
The sale was demanded by the European Commission after mergers and government support during the financial crisis gave Lloyd's excessive power in its market.
The Co-op had lost exclusive negotiating rights after doubts emerged about its ability to absorb the branches as part of Co-operative Bank.
Now the talks will resume on an exclusive basis as the two parties draw up a sale agreement and seek approval from regulators.
The sale, codenamed Project Verde, would hand the Co-op 5 million customers and a 7% share of UK current accounts.
Meanwhile, a rival bidder, NBNK, confirmed that it had made a revised offer for the Lloyds branches.
Look here to see how you'll be affected and here to see if your branch is to be transferred.