Induced-EMF due to the induced magnetic field of a wire?
A current carrying wire will always generate its magnetic field, if this is going to be opposite to the external field or not, depends in which direction the current flows.

That beeing said, there will be no induced voltage if there will be no flux change. For flux to change you can either keep the outer field constant and change the area (e.g. move the wire) or you keep the area constant and change the external field, or change the current in the wire, thus changing the total flux in the area.