Current—voltage characteristics of fabricated MoRe—Si(W)—MoRe heterojunctions are investigated in a wide range of changes in the parameters (such as thickness and dopant concentration) of the semiconductor Si (W) barriers. As found, at relatively high concentrations (5—9 at.%) of tungsten in the silicon barriers, a resonant tunnelling effect occurs through the localized levels of tungsten clusters. Simultaneously, under certain conditions, a superconducting Josephson current appears through such junctions and is conditioned by the multiple Andreev reflections of the Bogolyubov quasi-particles (quasi-electrons and quasi-holes) within them. The experimentally observed high quasi-particle excess current through the junctions indicates that there is a significant contribution of the Andreev reflections by the two $S/N$ interfaces into the charge transport through the studied Josephson heterostructures.