Its relay is probably got a welded contact, or its sensor is stuck in 'closed' circuit, as indicating "hot".
I would replace them both with new ones, as well as the thermostat.
Another condition is that the fan circuit is 'closing its contact' without the signal from the sensor, hence, turning itself on at the turn of the key. Have an electrician check it out; it's not so complicated.
This to be safe.