The Yorkshire Building Society (YBS) and Virgin Money have both written to the Competition and Markets Authority (CMA) to push for the big banks to carve out more new challengers from their existing networks of branches and customers.
Lloyds has already created TSB as a separate high street bank, and RBS is close to launching Williams and Glynn as a new, independent lender.
The exercise has taken both banks several years, and cost billions of pounds.
But the challengers say it is the best way to increase competition rapidly.
“The Big Five continue to hold an overly dominant market position, with younger customers effectively “sleepwalking” into opening an account with the Big Five,” said YBS, proposing “further forced divestitures”.
And Virgin Money, which is in the process of floating on the stock market, said: “It is worth examining the case for future divestments which could reduce concentration levels.”
They were joined by the business department’s former entrepreneur in residence, Lawrence Tomlinson, who said: “The only effective solution to the anti-competitive structure of the market” was to break up Lloyds and RBS “into their constituent parts”.
The big banks declined to comment further on the CMA’s investigation.