The process involves the sale of 760 million shares worth about $261 million to Kuwaiti nationals in the Bubiyan Islamic Bank.
Head of the founders' committee, Dirar al-Raba, said the Islamic bank has a capital of $330 million. Bubiyan is one of only two Islamic banks currently operating in the country.
The state-owned Kuwait Investment Authority (KIA) holds a 20% stake and the pension social security fund.
The capital is divided into one billion shares, each carrying a value of 33 cents. Subscription continues until 20 May.
KIA is the state investment arm that manages Kuwait's financial surplus mainly in foreign assets, but also has holdings in the domestic market.
These assets are estimated at $70 billion.
Until recently, the Kuwait Finance House was the only bank that had been operating in the emirate on Islamic principles.
It is one of the largest institutions in the region with total assets reaching $8.5 billion.
The Central Bank (CBK) has allowed a third establishment, Kuwait Real Estate Bank, to operate in accordance with sharia, or Islamic law, though it is still in the process of changing its status.
CBK has also withdrawn state guarantees for private sector deposits at local banks, valued at $35.6 billion, in a bid to boost performance and competition. The decision is effective from Monday.