The news, certain to fuel takeover speculation, came as suggestions of a link between Nationwide and Alliance & Leicester continued to grow.
Industry sources say the vote in favour of C&G's £1.8bn acquisition by Lloyds last Friday coupled with last week's approval by the High Court of the proposed Halifax/Leeds merger has turned the heat up on other talks within the industry.
A merger between Nationwide, set to be the UK's biggest society once Halifax converts to a bank, and third-placed Alliance & Leicester, would create an institution able to compete with the high street banks head- on, said analysts.
A spokeswoman for Alliance & Leicester would neither confirm nor deny that A&L was in merger talks: "We don't speculate on merger talks of this sort for obvious reasons. Until there is something more concrete we can't comment."
A spokesman for Nationwide said: "We are talking to a whole range of people and we don't want to comment on this particular rumour."
The A&L spokeswoman added that following the news from C&G and Halifax "everyone was turning to other parties. Everyone is under pressure to grow and consider all options."
Between them the defections of Halifax, Leeds and C&G to the banking sector have removed a third of the building society sector at a stroke.
A spokeswoman for C&G said that mergers would continue, and that C&G itself would carry on picking up "good, small, `Middle England' societies" after its takeover by Lloyds. The link-up will make Lloyds and C&G the fourth largest mortgage lender in the UK.
"We have merged with a lot of societies in the last seven years and we don't see the Lloyds deal putting a stop to that. We are looking for mortgage acquisitions. We know how to do it now.
In recent years C&G has merged with Guardian, which she said had low management expenses, as well as Bedford Crown and the Heart of England.
The rumoured A&L deal with Nationwide came in for criticism from analysts and other industry figures as making no sense, other than the sheer pursuit of size for the sake of it. One source said: "One has to question the idea, since the former mergers between the Alliance and the Leicester, and between Nationwide and Anglia, have not yet been completed."
Another commented that since the two societies were roughly the same size there would be a difficult squabble over who would take the top management positions.
A banker said that the possible Nationwide /A&L link-up looked to be driven "more by fears for the future than by strength".