There is not truth to this. How is there an American monopoly on money transfers?
Most countries have central banks that settle and controls its own currency.
Bigger banks have accounts at the central banks and can settle cross border without a problem.
The US has nothing to with it unless you are buying/selling USD (which, by the way, you can do outside of the US banking system, if you like) or you are buying selling some obscure currency which needs to be exchanged via the USD (or any other liquid currency for that matter).
Most countries have central banks that settle and controls its own currency. Bigger banks have accounts at the central banks and can settle cross border without a problem. The US has nothing to with it unless you are buying/selling USD (which, by the way, you can do outside of the US banking system, if you like) or you are buying selling some obscure currency which needs to be exchanged via the USD (or any other liquid currency for that matter).