you are going to have to buy something like this:
US Robotics Sportster External 33.6 Faxmodem Model 0459 unit only | eBay
But you are going to need to buy it from the country you live in because different countries often do not use the same sound frequency and cadence for the sounds we hear when dialing a phone number, such as ringback tone and busy tone.
As a result a US dialup modem is highly likely to NOT work when plugged into your telephone network in your country. And, if your telephone network does NOT provide a so-called “POTS” or “land line” then you are really going to be screwed.
Also you will be dialing internationally. There will be high long distance charges.
There are NO services that I know of that let you virtually send bytes to a specific US telephone number over dialup.
What you really need is to find someone living in the United States preferably in the state that the destination telephone number is located in (unless it’s an 800 number) who is willing to setup a machine with a modem on it you can SSH into and develop on and buy a telephone line that plugs into that modem. Any “old school” ISP could do this - if it was back in 2004, 20 years ago when I was working at my old ISP employer Internet Partners - this would be a snap, we’d write up a contract and build up a physical machine for you, get a POTS phone line, plug it in and you could then SSH into it and go to town. We had a pile of modems so no problem there.
Today, you are going to have as much problem finding someone in the US who can do this as you are having figuring it out, anyone under the age of 45 probably has zero experience with any of this tech. This is really, really old school. Plus the fact you want to dial into a bank/payment terminal processing facility is already going to freak a lot of people out.
It sounds to me like you got a contract from someone in the US wanting cheap telephony development done who doesn’t know the first thing about the Public Switched Telephone Network. They probably talked to a bunch of phone devs in the US my age who quoted them a LOT and went overseas. Is there any way you can go back to them and say “look if you want me to do this your going to have to do some hardware setup in the US”
A couple of other things you should probably know as well:
Just about all these dialup processing networks limit to either 300baud or 1200baud because the higher speeds are unreliable over the PSTN. And a LOT of the “more modern” 56k modems have serious problems with that low of a speed. I’m PRETTY sure I’ve done 1200Baud connections with that 33k Sportster back in the day. But, it’s going to be something of a crapshoot. You will have to force the serial port on the PC into 1200bps to make the modem do 1200baud. That also requires a REAL serial port in the PC not one of those USB-to-serial port “dongles” The dongles often drop characters and other such rubbish.
The Linux drivers do best with the MOSChip based serial ports, just as a FYI.
Incidentally I still, to this day, use a modem. I use it to dial my cell phone for paging alerts, the reason is that if my Internet connection dies then it wouldn’t normally be able to send me an alert so the PSTN is the only way. It’s plugged into a real POTS line and a FreeBSD system. It’s actually a voicemodem and when it makes a call to my phone I answer it, like it’s a phone call, and it sends out a prerecorded warning alert. The reason I have to do it that way is because all of the cell companies have shut off their dialup paging gateways.