Source: Sharecast
According to Sky News, Aldermore - which had hired advisers to work on a takeover of its rival mid-sized lender - opted not to submit an offer ahead of a deadline earlier this month.
Its withdrawal will come as a surprise, and will raise questions about its future strategy under Steven Cooper, its respected chief executive, Sky said.
The decision leaves Shawbrook as the only known bidder for the Co-operative Bank, which has been put up for sale by the syndicate of financial investors which own it.
Analysts had named Aldermore as the most logical strategic buyer of the Co-operative Bank, although City sources told Sky on Wednesday that a number of other parties remained interested in bidding.
Aldermore's plans are said to have been derailed by a leadership shake-up at FirstRand, the South African banking group which owns it.
Other lenders cited as potential bidders, including Paragon Banking Group and OneSavings Bank - are also yet to table offers.
Shawbrook, another medium-sized British bank, has approached the Co-operative Bank's advisers with a £3.5bn merger proposal.
That largely consists of Shawbrook shares, however, and it was unclear how attractive that offer would be to Co-op Bank shareholders, according to Sky sources.