There are a few options available for making direct bank-to-bank transfers without the use of a bank API. One solution is to use a payment processor or fintech company that specializes in bank transfers. Examples include TransferWise, Payoneer, and Stripe. These companies have built their own payment infrastructure and can facilitate bank transfers without relying on a bank API.
To implement this, you would need to integrate with their API and provide the necessary information, such as recipient bank details and transfer amount. The payment processor would then handle the transfer on your behalf, usually for a fee.
It's important to note that there may be restrictions and limitations on the amount and frequency of transfers, as well as the countries and banks supported, so be sure to carefully review the documentation and terms of service for each provider.