The scanner sends ultrasounds to detect 3D ridges of fingerprints in order to recognise users.
Samsung said it was "aware of the case of S10's malfunctioning fingerprint recognition and will soon issue a software patch".
South Korea's online-only KaKao Bank told customers to switch off the fingerprint-recognition option to log in to its services until the issue was fixed.
Previous reports suggested some screen protectors were incompatible with Samsung's reader because they left a small air gap that interfered with the scanning.
The British couple who discovered the security issue told the Sun newspaper it was a "real concern".
After buying a £2.70 gel screen protector on eBay, Lisa Neilson registered her right thumbprint and then found her left thumbprint, which was not registered, could also unlock the phone.
She then asked her husband to try and both his thumbs also unlocked it.
And when the screen protector was added to another relative's phone, the same thing happened.
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