Traditionally, hand-to-hand currency has a zero nominal interest rate. Those holding it sacrifice nominal interest earnings. Those issuing it can borrow at a zero nominal interest rate.
For the most part, that was also true when competing private banks issued hand-to-hand currency. Still, any profit from borrowing at a zero nominal interest rate was likely to be competed away. For a comprehensive bank, issuing both deposits and currency, higher interest rates on deposits would be an obvious avenue for such competition. Another avenue of competition would be lower interest rates charged on bank loans.
Along with those sorts of price competition, there could be efforts to make it easier to obtain currency. For example,a bank could provide more branch offices, more teller windows, and longer hours. Under modern conditions, more ATM machines would be a strategy directly aimed at making it more convenient to obtain and launch into circulation a particular bank's currency.
If there were specialized "banks of issue," that solely issued currency and invested the proceeds in uniform "bonds" that were also directly held by other investors, then competition by paying higher deposit interest rates or charging lower rates on specialized bank loans would be impossible If the currency was all launched into circulation by the specialized banks of issue purchasing bonds (much like an open market operation,) and the sellers of the bonds then deposited the currency into ordinary deposit banks, then more branch offices and the like would also be irrelevant. Perhaps it is natural to assume that such banks could only compete by paying higher prices for the uniform "bonds," causing "the" interest rate to be too low with some kind of adverse inflationary consequence.
However, unless there are regulations forbidding it, it would seem that banks also making deposits and loans would find it more attractive to issue their own currency and capture the profits from borrowing at a zero nominal interest rate. And then, it is easy to see how they would compete away any excess profit.
Interestingly, this would make it very difficult for any bank to operate solely by issuing deposits and making loans. Under competition, the "excess profit" from issuing currency would be necessary for the banks to cover operating costs. The benefits would go to bank customers.
However, with modern data processing technology, it would be possible for banks to pay interest on currency to whomever launches it into circulation. When currency is withdrawn by a customer from an ATM machine or from a bank teller, the particular notes would need to be identified. If the currency was withdrawn from a deposit account, that deposit account could continue to be credited with interest. When the currency is returned to the issuing bank, then the currency would need to be identified electronically, and then interest would no longer be credited to the deposit account. If the currency was withdrawn against a line of credit, then the interest rate charged on the loan could be reduced as well, perhaps to nothing, until the currency is spent, deposited in some other bank, and cleared.
If a depositor withdraws currency from a bank to hold, then he would continue to pay interest. There would be no opportunity cost for holding currency, unless the interest rate paid was lower than that on deposits, reflecting perhaps printing costs. Once the currency is spent, then the interest continues to be paid to the person who launched it into circulation. Those receiving currency in payment would obtain no interest. Of course, if they find that bothersome, they can simply deposit the currency in their own bank. They could leave it in an interest bearing deposit or withdraw currency from the account. Then the person receiving and holding the currency would earn the interest.
Suppose the interest rates that banks can earn fall so much that the competitive interest rate on currency becomes negative. Interestingly, there is no reason why banks could not issue currency and charge interest to the deposit accounts from which the currency was withdrawn. The problem would be that whoever withdrew it suffer persistent costs if they spend it and launched it into circulation. Those receiving it might choose to hold it, or it make change hands among others. In a scenario where the competitive yield on currency is negative, those receiving currency in payment would have an unusually strong incentive to hold it or perhaps use it to make payments. Regardless, whoever launched the currency into circulation would continue to pay the cost.
This suggests that if the zero nominal bound is approached, and issue of currency becomes costly, then it would be the banks' own customers that would stop using currency and instead switch to payments by deposit. Further, those making the decision to withdraw currency and use it would have a strong incentive to not only accept, but insist on call provisions for hand-to-hand currency.
It is obvious that a period when issuing currency is not very profitable would not be an opportune time for a bank to instroduce its own interest-bearing currency. But if nominal return on banks' earning assets become sufficiently high, and government regulation can be avoided, it should be possible to introduce interest-bearing currency.