If a merger goes ahead it would create a combined group with assets over £35bn.
The news comes as part of a strategic review at Chelsea at the start of which its new chief executive Stuart Bernau said he would look at all options for the society including possible mergers.
In August, Chelsea posted a £26m loss for the first half after a £41m mortgage fraud. Its previous chief executive Richard Hornbrook and finance director Andrew Parsons quit the same month, although the society insisted this had nothing to do with the losses.
In March, Chelsea posted £29.2m losses due mainly to £55m exposure to the Icelandic banking collapse, but the mutual says it has now exited its Kaupthing Singer & Friedlander position with a £9m gain compared to its provision.