Answer: The short answer is: Probably.
The long answer would be: That depends.
The short answer would indicate that any number of factors is involved such as: whether a person is holding a large quantity of coins (possibly for retirement) and/or whether a person is buying coins and/or other types of paper bills (which may be considered “currency”).
The long answer would indicate that banks can do just about anything they see fit. For example, there are various laws in some U.S. states that make it a crime to buy a lot of coins, coins that are “obsolete” (they have ceased to be in the U.S.) or coins that are “worth less than the market value of the coin,” or any coins in a pile too low to be called a “monetary unit.”
On the topic of buying coins to cash in, there are numerous sites listed above that may sell you a good deal for dollars with no coins in it that someone is interested in selling to you on their site.
In our opinion, most of these sites are not banks so we could not confirm that you could cash in their coins on their site. It just is a possibility.
The second point about banks selling in this way is that they are prohibited by the United States Postal Service and by almost all other postal services from directly accepting coin payments.
Instead, banks are prohibited from directly accepting any kind of bank wire transaction.
On the topic of buying coins in bulk for cash, there is an alternative. The vast majority of the United States coin-purchasing people that we talk to use these sites to place bulk purchases of coins and that, in general, it is easier to pick up coins for cash.
Banks and other depositories can do anything on that site even in the event that one is not actually in the United States. If they sell you coins for cash on a certain day, they will probably send you a notice. They may also ask you if the “cash equivalent” in coins were available to sell (remember the question to the banks). That is, they may request the same info from you they use for cash at the banks.
The best rule of thumb to follow is that banks can generally, though possibly not always do any “cash in” purchases of coins or other forms of paper bills because they have so many rules in place that restrict them from taking money for that.