The executive board of TSB has agreed to accept a £1.7bn ($2.5bn, €2.24bn) takeover offer from Spanish bank Banco de Sabadell.
Under the agreement the Spanish lender will initially buy 9.99% of the stake in TSB held by Lloyds Banking Group; this part of the deal will go through on March 24 and will be priced at £170m.
Lloyds has also entered an "irrevocable undertaking" to accept the offer with regard to its remaining 40.01% stake in TSB, the FT reported.
TSB shareholders will receive 340p for each share. Since the deal was tabled earlier this month the shares have gone up 15%.
Sabadell said it intends to "continue to operate TSB as a robust competitor in the UK banking market, building on the TSB brand name".
Oliu Creus, chairman of Sabadell, said: "We believe that our experience of growing SME lending, our resilient and tested IT platform and our commitment to innovation will speed up TSB's expansion so that it fulfils its potential as a strong and effective challenger to the traditional UK banks, without any of their legacy issues.
Lloyds said the expected £850m from the sale of its TSB shares would be used for "general corporate purposes".
António Horta-Osório, group chief executive of Lloyds Banking Group, said: I am delighted to confirm we have agreed terms for the sale of our remaining stake in TSB to Sabadell. This is a significant and positive step for the Group and will enable us to meet our commitments to the European Commission, well ahead of its mandated deadline.