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TPG had been lined up to pay £179m for a 23 per cent stake in the bank. The move was prompted after the credit rating agency Moody's announced it was downgrading the debt of Bradford & Bingley.
However, the group's decision to pull out of the deal has prompted Bradford & Bingley to increase the amount it needs to raise from its investors in a cash call from the £258m amount previously stated to £400m.
Legal & General, Standard Life, M&G and Insight Investment are understood to be backing the enlarged rights issue at a subscription price of 55 pence.
The decision by TPG follows entrepreneur Clive Cowdery's investment vehicle Resolution's decision last week to also scrap its proposal to invest £400m in the lender in exchange for a controlling stake in the bank.
At the time, Resolution blamed the "the entrenched position of the board of Bradford & Bingley".