You can make the coil part of an oscillator and measure the frequency, add a capacitor so this arrangement works at the resonant frequency. Use a resistor in the ground connector of the oscillator to measure the current this oscillator takes by measuring the voltage across this resistor..
When the core is introduced into the coil the resonant frequency of the system will change and so will the current the oscillator takes.
Alternatively use an IR LED and photo transistor and mount it so the moving of the core breakers the beam.
Alternatively mount a magnet on the end of the core and have a reed switch or Hall effect switch to detect when the magnet comes close.