We investigate the effects of the magnetic field on the propagation of laminar flames of nuclear reactions taking place in white dwarfs with masses close to the Chandrasekhar limit. We calculate the velocities of laminar flames parallel and perpendicular to uniform magnetic fields as eigenvalues of steady solutions for magnetic hydrodynamical equations. As a result, we find that even when the magnetic pressure does not dominate the entire pressure it is possible for the magnetic field to suppress the flame propagation through the thermal conduction. Above the critical magnetic field, the flame velocity decreases with increasing magnetic field strength as v ∼ B −1 . In media with densities of 10 7 , 10 8 , and 10 9 g cm −3 , the critical magnetic fields are orders of ∼10 10 , 10 11 , and 10 12 G, respectively.