The only example I'm thinking of is in the biblical book of Esther, set in the Persian empire, someone offers to give the king 10,000 talents of silver "in the hands of the people who do the work" which probably means "the people who do the work of transporting all that silver into the treasury." So at least from that, it sounds like there were people whose job was to physically move the king's money around

In both Ancient Rome and Ancient Greece they'd often just go through agents who would act as bankers. A Roman could deposit his money with a banker (called argentarii among other names). And when he'd need to pay someone a large amount of money he could just give them a Roman equivalent of a check (perscriptio) and the bank would pay it out. The recipient of the money could get physical money, create an account and have it held in his name or perhaps already had an account there.

Banks also lent money at interest if you wanted to make a large transaction without all of the money at hand. And bankers could act as mediators in complex transactions. So you could have multiple banks in multiple cities working with each other to put together a complex financial transaction on your behalf. Banks kept very detailed records, some of which have been found in places like Pompeii.

Obviously it depended where you were stationed in the Roman empire. Places not served by bankers would have to make physical transfer of currency or set up their own arrangement for payment in installments. But most major cities would have them. And bankers could arrange transactions between themselves.

I'm more comfortable with Ancient Rome but you should definitely keep in mind that it wasn't a Roman invention. The Greeks had this too. And that's just what I know about. I'd be surprised if the Phoenicians didn't have some form of banking.