In the tight-binding approximation, we take into account the next-nearest-neighbor hopping in graphene that leads to nonrelativistic-like corrections in its low energy spectrum. The electronic density of states in a magnetic field is found and the fan diagram is plotted, which interpolates between those for the relativistic and nonrelativistic limiting cases. It is shown that the Berry phase for the system under consideration coincides exactly with its value for the relativistic system.