It could. A changing magnetic field implies that there exists a changing electric field. If the electric field is aligned with a parallel component (which may or may not happen since we only know that the wire is normal to the B field) then it will induce a current in the wire.
Now lets say the wire is shaped into a square loop with the center coinciding with the center of the changing B field(again, B field perpendicular to square).Taking circular amperian loop with radius s from the center of the magnetic field, the Induced E field can be found to be
E=-s/2. dB/dt.[tex]\phi\widehat{}[/tex]. Assuming the B field is uniform throughout the plane, and is changing at constant rate. So, if the square loop is put at a distance from the center of the field, then the induced E field in it seems to be different as E has the dependence on s. Thus, result in a seemingly difference emf induced if the loop is placed at different region.
However, from faraday's law, [tex]\epsilon[/tex]=-d[tex]\Phi[/tex]/dt. Since the area enclosed by the loop is the same plus the rate of change of B field is the same, it implies that the emf induced is the same in the loop no matter where it is placed in the B field.