The acquisition would be part of a £400m rescue package for the struggling bank, according to Sky News.
Co-op has also attracted interest from many other private equity suitors including Apollo Management.
Corsair, whose partners include former trade minister Lord Davies, is looking to invest tens of millions of pounds into the business, said Sky.
The Co-op Group’s stake in the bank has declined after a £1.5bn hole was discovered in its balance sheet last year. The group has ceded control of the business to bondholders including hedge funds Perry Capital and Silver Point.
Co-op currently holds a 30 per cent interest in the bank and is unlikely to invest sufficient funds to keep its stake from falling below 20 per cent.
Last November Co-op turned down bids for its funeral care unit from private equity suitors including CVC and Montagu.
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