Core saturation always causes primary current to INCREASE due to the fact that the self inductance goes away after the point of saturation. The load limits the current. So a saturated core causes the inductive reactance of the primary to drop. Its already low since the idea of a current transformer is to avoid adding any extra impedance in series with the load. Ohms law says the voltage across the primary should drop if there was any voltage there to begin with. I would agree with phrak in that there could be some distortion at crossover since there will be a small part of the cycle around zero current (crossover) that there is enough self inductance to keep things 'normal' at these low currents. However, as soon as the core saturates, the impedance of the primary circuit drops to zero (where it should be) and everything goes on as normal other than a fried current transformer core and possibly secondary winding. I would say if you want to prove this wire a CT in series with a resistor and hook it up to a variac (preferrably isolated). Then using a scope watch the voltage across the resistor as you slowly increase the variac output voltage. You might have to experiment with a slightly loaded secondary in order to prevent the core from saturating in the too-low-to-measure region.
I wish I had some CTs to experiment with, however, I don't. The best I could probably do is to use a small step down VT and try sneaking a turn of wire around the core and using the primary as the 'secondary' of my newly created CT. My single turn of wire should present a pretty low impedance so it would probably work fine as a CT for experimental purposes.