Molecular dynamics simulations are performed for a polymer melt composed of short chains in quiescent and sheared conditions in supercooled states. In quiescent states, while the self chain motions are excellently described by the Rouse model, the stress relaxation function G(t) exhibits a stretched exponential decay in a relatively early stage and ultimately follows the Rouse function. Transient stress evolution after application of shear obeys the linear growth integral(0)(t) dt'G(t') for strain less than 0.1 and then saturates into a non-Newtonian viscosity. A glassy component of the stress is much enhanced as compared to the entropic component.