That's sort of the idea behind securitized mortgages, except once the bank makes the loan the bank leaves the picture (except perhaps as a custodian responsible for payment collection sometimes.) This is one reason securitization of mortgages led to lower interest rates and was generally considered a good thing for a while.
In the alternative scenario, economists describe loaning from demand deposits (at a higher interest rate) with a phrase like "the bank is selling liquidity".
