There definitely are magnetic fields around the current in a wire. This is why it is called electro-magnetism. The two go together. Whenever a magnetic field crosses a wire it induces a current in that wire. This is how electricity is generated in the first place, transmitted long distances at high voltage then stepped down with transformers, how cartridges generate a signal, on and on. This is also how antennas work, including the one in your cell phone. This is why RFI is such a problem. On and on.
This used to be the kind of thing they taught in school. Grade school. We would put iron filings on a sheet of paper, hold a magnet under it, watch the filings line up along the magnetic field lines. Then wrap a wire around a nail, attach to a battery, see the same thing with this electro-magnet.
They didn't teach you this in school?