The dynamical response of entangled polymers can be understood in the framework of the tube theory. In this paper, the dynamics of linear entangled polymers that reptate under the additional action of drift along the primitive path, are explored. A modified reptation model with drift, valid for small drift values, is presented and solved analytically, as well as by means of Brownian dynamics simulations. When the drift mechanism dominates over diffusion, the transport properties of these active entangled chains change dramatically with respect to the pure reptation case: the viscosity grows linearly with the molecular weight whereas the self-diffusion coefficient becomes independent of the molecular weight. Some open questions regarding possible ways to generate a drift such as the one described are discussed briefly.