We propose a degenerate V-type inhomogeneously broadening atomic model for Kerr nonlinear control under the support of an external magnetic field. The magnetic field is used as a control knob to convert the medium from transparency to absorption at the resonance center. It is shown that the magnitude and sign of the Kerr nonlinearity index are manipulated by the magnetic field. Especially, we can easily convert the Kerr nonlinearity index from the positive peak into a negative peak, and from zero value can be transferred to enhanced positive or negative peaks simply only by changing the direction or strength of the magnetic field. In addition, the Kerr nonlinearity index under the influence of the temperature, the intensity, and detuning of the control field is also considered. The proposed model can be useful for applications in optical switching and nonlinear optical devices as well as possible experimental implementations.