A short circuit is usually taken to be a fault that reduces the voltage to near zero volts. How much that will increase the current will depend on what the power supply is.
Are you going to tell us?
Some power supplies have "fold back current limiting", where they produce less current when shorted than into their normal load. I had problems with that were there was a 2.5 A power supply, and the circuits were fused at 2 A. A short circuit resulting in the power supply only producing around 0.5 A and the fuse didn't blow.
If you have a power supply like that, current measurement won't help, so please say what the power supply is, and what it is that you are trying to protect.