TSB said it had been offered 340p in cash per share, about a 30 per cent premium on last night's closing price, valuing the lender at £1.7bn. The board of the bank said it is "willing to recommend an offer at the proposed priced", subject to it reaching an agreement on other terms and conditions.
This morning a statement by Lloyds said it "would be minded to accept an offer at this price if it is made".
If the deal is accepted, it would be more than twice the £750m offered by the Co-operative when it came close to buying the bank in 2012. After several weeks of talks, the Co-op suddenly pulled out, a move described as "bewildering".
However, it was later regarded as a lucky escape for TSB: months later, the Co-op Bank uncovered a £1.5bn black hole in its balance sheet.
Shares in TSB were up 25.9 per cent in late morning trading, at 332.5p.
Since it was spun-off from Lloyds in September 2013, TSB has positioned itself as a "challenger bank", focusing entirely on its retail offering. Chief executive Paul Pester has focused on retail and small business lending, shying away from launching an investment operation.
Today the lender said Sabadell could "support and accelerate TSB's retail growth strategy and accelerate the expansion of TSB's presence in the [small business] sector".
The lender launched an IPO in June last year, with shares priced at 260p, which then jumped 10 per cent on the first day of trading. In February it posted figures showing profits had doubled in 2014.