The R(T,I) shape of the superconducting transition in Transition Edge Sensors (TES) is of crucial importance to determine their ultimate performance. This paper reports a study of the temperature and current dependences of the transition of Mo/Au TESs, focused on the low resistance region, where these devices preferentially operate. A large broadening of the transition is observed when increasing the applied current. An empirical analytic expression for R(T,I) is found, which describes the transition of devices with different critical temperatures, from R=0 up to at least 30% Rn (in some cases nearly 80% Rn). Several mechanisms for this behaviour are considered; results show that a current assisted vortex pair unbinding mechanism (Berezinskii-Kosterlitz-Thouless transition) could be the possible origin for this behaviour. Finally, the consequences of the current-induced transition broadening for TES properties and operation are outlined.