The Resolution offer was part of ambitious plans to consolidate smaller banks as one larger and stronger institution.
In a statement to the stock market B&B says it received the funding proposal on Friday, but after consideration it has decided it will not recommend the proposal to shareholders.
The bank is continuing to support the restructured rights issue and TPG Investment previously outlined and will be recommending this to its shareholders.
Resolution had offered to pump £400m into Bradford & Bingley, which would have replaced the current rights issue and seen the company take effective control of the bank.
Resolution expects its consolidation drive to require a £2bn investment over the next two years in order to create one institution from up to five smaller banks, with a balanced business model and an emphasis on generating retail savings.
Cowdery is now expected to launch a campaign in order to generate shareholder support for the Resolution proposals.
He was invited to lead an alternative proposal to rival rights issue and the TPG investment by some of the banks existing shareholders, reportedly Standard Life, Legal & General, Prudential and Insight.