We measured the time-resolved transmission of picosecond terahertz pulses through a modulation-doped Al(0.3)Ga(0.7)As/GaAs heterojunction in the presence of a D.C. current. The introduction of a current changes the transmission on the order of one percent and causes additional dispersion. This effect can be explained by a drift-velocity-shifted Fermi distribution in the Lindhard formula for an electron gas. The effect is especially pronounced in high-mobility electron layers in semiconductors, where the momentum relaxation time is long, and modest currents result in large drift velocities. In our degenerate 2D samples, within attainable regimes of drift velocities below the Fermi velocity, the amplitude of the effect grows with the current. The current-related effect can be separated from background effects due to a marked polarisation dependence. The experimental results are well matched by the theory based on the aforementioned model and by using the self-consistent solution of the Maxwell equations.