Monte Carlo simulations are reported for the isotropic to nematic phase transition in semiflexible polymer melts. The molecules are modelled as tangent hard sphere chains with a bending energy that can be used to tune the stiffness of the chains. In the rod limit an isotropic to nematic transition occurs at high pressures. For freely-jointed chains no nematic phase is found, but a fairly sharp transition to a nematic phase is observed when the stiffness is increased at fixed density for sufficiently high densities. In semiflexible chains the isotropic to nematic transition is accompanied by a discontinuous change in the chain dimensions. The simulations show that existing theories are not quantitatively accurate for the isotropic to nematic transition in semiflexible chains.