TSB said Sabadell has made a proposal of 340 pence in cash for each TSB share, and that talks were continuing.
"The Board of TSB has indicated to Sabadell that it would be willing to recommend an offer at the proposed price, subject to reaching agreement on the other terms and conditions of any offer," TSB said in a statement.
Sabadell, Spain's fifth-biggest bank, is planning to diversify by expanding overseas to offset sluggish growth in its home market, which is in the early stages of recovery following a deep recession.
TSB, Britain's seventh-biggest lender, said it believed Sabadell could support and accelerate TSB's growth in personal banking and help it expand its small business lending.
TSB listed on the London Stock Exchange in June last year in an initial public offering (IPO) which valued the business at 1.3 billion pounds or 280 pence per share.
Lloyds was ordered to sell the business by European regulators as a condition of its bailout during the financial crisis of 2007-9. It still has a 50 percent stake in the business.