Santander and TSB have both ruled themselves out of bidding for the Co-operative Bank, according to reports.
Spanish bank Banco Sabadell was thought to be interested in bidding through TSB, according to the Financial Times.
Sabadell bought TSB from Lloyds Banking Group in 2015.
However, Sabadell chairman Josep Oliu has ruled out further UK acquisitions in an interview with Spanish newspaper Cinco Días.
Santander has decided the Co-op is not large enough to help its UK growth plans, according to sources close to the auction.
The British lender announced it was up for sale last month.
The Co-op announced 2016 losses of $477m earlier this month.