Putting a tool in a powered coil will magnetize it temporarily but I don’t think it will stay magnetic or at least not for long. They sell little magnetizers that you can get cheap that will last a lot longer though if that’s what you’re looking for…
I have one like this around somewhere that worked well when I tried it.
When I was a kid my Dad made a magnet charger with some electrified copper coils would this be something that could be used to achieve similar results somehow? TIA (i.imgur.com)