An electric current will start to form when a part of the wire is within a magnetic field. Using the magnet you could put the whole wire into the magnetic field of the magnet and move it across the magnet so that a current starts to run.
Another way you could do it is to make the wire so that it looks like a coil and stick the magnet in the middle of the coil of wire. This will cause an induced current in the coil of wire.
