Measure DC current with current transformer

If you terminate your current transformer with a resistor to get a voltage signal out, a normal AC current transformer would saturate very quickly if you apply any DC current due to Vt=NAB where V is positive value (or negative, same effect) and as t goes on, B just keeps building until saturation. For AC, V keeps changing and nets out to zero.

In a zero flux current transformer, you use an amplifier to sense your output and drive a current in a secondary coil, wound in the opposite direction (or apply an inverse signal, take your pick) to cancel out the flux build up. This results in a new operating point where equilibrium is reached and the core won't saturate yet you can measure the current as a DC voltage which is proportional to your applied current in the secondary coil.

Here is an image borrowed from Hioki, a manifacturer of such modules:

It's not possible to measure a DC current with a transformer. In the case of AC the changing magnetic field associated with the current in the primary induces a current in the secondary winding which is what you can measure.

As mentioned in the comments above if you need to detect a DC current you can use a Hall effect device which does not require a changing magnetic field see https://en.wikipedia.org/wiki/Hall_effect for a bit more detail.