TSB announced that the Spanish bank Sabadell has approached the company for a £1.7bn takeover.
The news sent TSB shares soaring by more than 25%.
Less than a year ago TSB rejoined the stock market after Lloyds sold off a 50% stake in the business.
Lloyds has been ordered to sell the bank by regulators as a condition of its 20 billion pound ($30 billion) government rescue during the financial crisis of 2007 to 2009.
In a statement Lloyds said it “would be minded to accept an offer at this [340p] price if it is made”.
Sabadell has until 9 April to make an offer.
Lloyds, which is still 23-percent-owned by the government, sold off its Spanish retail banking business in 2013 for a 1.8% stake in Sabadell.
Sabadell is Spain’s fifth-biggest bank. It made a proposal of 340 pence in cash for each TSB share.
The Spanish bank plans on diversifying and, due to sluggish growth in its home market, expand its presence overseas.
Sabadell will “operate TSB as a robust competitor in the UK banking market”
TSB said that if the takeover goes ahead then Sabadell will “operate TSB as a robust competitor in the UK banking market, building on the TSB brand name”.
Adding that Sabadell could help accelerate its “retail growth strategy”.
The offer is 29% higher than TSB’s closing price on Wednesday.
The deal is almost nearly twice as much as what it would have made out of the original plan to sell the business to the Co-operative Bank – a deal that eventually collapsed.
It was after the Co-op deal collapsed that Lloyds floated TSB on the London Stock Exchange.
