OSB shareholders would own about 55% of the combined company and Charter Court shareholders would own the rest.
OSB chief executive Andy Golding would become the boss of the merged company.
Banks say merger has 'compelling strategic and financial rationale'
In a joint statement, the banks said they believe the deal has the potential to create material shareholder value and has a “compelling strategic and financial rationale”.
They said the deal would create a leading specialised mortgage lender in the UK with greater scale and resources to take advantage of growth opportunities.
OSB was founded in 2011 as part of a private equity buyout of the Kent building society KRBS.
Specialist mortgage bank, Charter Court, was launched in 2008.
In morning trading shares in OSB gained 9.5% to 405.4p and Charter Court rose 9.1% to 334.2p.
Deal terms 'reasonably sensible', says ShoreCap
Shore Capital maintained a 'buy' rating on both stocks, saying: "Overall, we view this proposed combination as relatively unsurprising given the strong similarities between the two organisations. The terms appear reasonably sensible to us given that we currently see similar upside to fair value for both stocks (of around 50%).
"Execution risk should also be relatively low, in our view, given the similarities between the two organisations and the fact that the deal appears to have been proposed on friendly terms.
"We see the likelihood of an interloper or alternative bidder for either company appearing as being low."
The banks were among those created following the financial crisis with the aim of challenging Britain’s biggest lenders including Lloyds Banking Group PLC (LON:LLOY), Barclays PLC (LON:BARC), HSBC Holdings PLC (LON:HSBA) and Royal Bank of Scotland Group PLC (LON:RBS).
However, consolidation of the challenger bank sector started with the £1.7bn merger of CYBG PLC (LON:CYBG) and Virgin Money Holdings PLC last October as tough competition and cost pressures have put a squeeze on smaller British banks.