Brazilian bank Safra and Chinese rival ICBC are also bidding for BSI, two of the people said.

Generali is selling BSI as part of a plan to free up 4 billion euros of capital from non-core assets and meet new rules for European insurers.

Bankinter, Apollo, Generali and BSI declined to comment. Safra and ICBC were not immediately available for comment.

"It's a tough deal", said one of the people, who was speaking on condition of anonymity because the talks are private.

"BSI is a good asset. The problem is that it is big and has a very restricted number of possible buyers."

Generali, Europe's third-largest insurer, bought BSI just before the financial crisis and has been trying to sell it for more than two years.

Price had so far been the main sticking point since potential buyers believed BSI was worth less than its book value of 2.3 billion Swiss francs (1.6 billion pounds), two people working for possible suitors previously told Reuters.

A rise in political risk following the inconclusive Italian election could also weigh on the process, one of the people said, although BSI has been seeking to diversify operations by investing in Asia.

Japanese firms Mizuho Bank Ltd and Sumitomo Corp, a Chinese private equity fund and Middle Eastern players in Qatar and Abu Dhabi have also shown interest in BSI, but have not made offers so far, one of the sources said.

Bankinter is one of Spain's healthier banks, with less exposure than others to a damaging property boom and bust, which ultimately led Spain to ask Europe for a 40 billion euro bank bailout.

(Reporting by Sophie Sassard; Additional reporting by Jesus Aguado in Madrid; Editing by Mark Potter and Jane Merriman)

By Sophie Sassard

Stocks treated in this article : Sumitomo Corp, Bankinter SA, Assicurazioni Generali SpA, SAFRAN