A senior source said: “Nationwide has its eye on West Bromwich and Britannia but doesn’t know if it would take them until it is offered the chance to look at the books.”

Nationwide, which has already agreed to buy the Derbyshire and Cheshire Building Societies, declined to comment.

Consolidation is one way of dealing with the financial turmoil that is weighing heavily on parts of the sector.

Nationwide dwarfs many of its competitors and is understood to have been a beneficiary of the downturn in the financing and mortgage markets in recent months, because savers have deserted banks and smaller building societies in a “flight to safety”.

When Nationwide reported results for the year to April 4, chief executive Graham Beale unveiled a £9.1billion increase in net deposits, and a market share of about 19 per cent.

Many smaller building societies with five to 10 branches are facing fierce competition from savings rates at bigger banks, and could be snapped up.