A deal to sell 632 branches to the Co-operative Group is still on track, Lloyds Banking Group said on Wednesday.
There had been fears the sale had stalled because of concerns about a deal which would triple the size of the Co-op's banking arm.
But part-nationalised Lloyds said it had made "considerable progress" with the Co-op and was now in exclusive talks with the group.
Lloyds has to offload the branch estate by November 2013 as a penalty for receiving State aid at the height of the financial crisis.
The branch business accounts for a 4.6% share of the UK current account market and up to 19% of Lloyds' mortgage book, with around five million customers.
The Treasury said it welcomed Wednesday's announcement by Lloyds, which it said was a "significant milestone" in efforts to improve competition in the banking sector.
A Treasury spokesman said: "Although the deal has not been finalised, we warmly welcome this development as a positive step in the process of delivering the Lloyds divestment, and the benefits that will have for competition and the mutual's sector."
Lloyds chose the Co-op as its preferred bidder in December, but sale plans have already suffered lengthy delays and initial hopes to sign a deal by the end of March were dashed due to protracted talks with regulators.
Union Unite said the delays were putting staff under pressure.
David Fleming, Unite national officer, said: "Months of speculation and negotiations have left the Lloyds banking workforce feeling vulnerable."
There has been mounting speculation that Lloyds was struggling to secure the sale - known as Project Verde - and that it may have have to slash the sale price by as much as 50% or float the business.
Lloyds, which is 40% owned by the Government, gave no details on the sale price in Wednesday's statement, but said it had reached an "understanding on the commercial terms for the transaction".
The decision to hold exclusive talks with the Co-op means rival bidder NBNK is no longer in the running for the Verde estate.
NBNK had hoped to step in if The Co-op deal failed, but confirmed it was now no longer in pursuit of the branches.
Gary Hoffman, NBNK chief executive, said: "We are disappointed that the door has now been closed on this opportunity."
It is thought that some of the regulatory uncertainty surrounding the Co-op offer has been addressed through plans for Verde's management team to transfer with the business in order to run the Co-op's enlarged banking arm.
Lloyds is also said to be providing the systems and technology platforms needed to run a large banking operation.
However, there is still concern that because the financial services arm will dwarf the rest of the Co-op business the whole of the Co-op will have to be regulated as a financial institution.
It is not clear whether monitoring of this kind will meet with the approval of the Co-op board, which is a group of elected member representatives and customers.