The merger between the Yorkshire, the second biggest society in the UK and Chelsea, the fifth biggest, would create a firm with nearly 3 million customers and a potential rival to market leader Nationwide.
In a joint statement, the two say the combined entity, which would have assets of £35bn, “has the potential to create a second major force in the building society sector”.
The Yorkshire has 143 branches, while Chelsea has 35.
Chelsea, which has been struggling and reported a loss of £39m last year, says the proposed merger follows a review of its “activities, operations, financial position and corporate structure”.
The building society sector has seen a flood of merger activity in the last 18 months.
The Nationwide took over the Cheshire and Derbyshire building societies last September and then rescued the Dunfermline building society in March, while the Yorkshire acquired the Barnsley building society.
Co-operative Financial Services and the Britannia Building Society completed their merger in August.
The Yorkshire and Chelsea merger will be subject to approval from their members and the Financial Services Authority.