Tungsten (W) samples containing 0, 1, 3, and 5 at.% rhenium (Re) were irradiated by 20 MeV W ions to 0.5 dpa at 290 K and to 0.3 dpa at 1350 K. An additional set of samples was irradiated to 0.5 dpa at 290 K and then annealed at 1350 K. The irradiated samples were exposed to a deuterium (D) plasma at 370 K. D concentration profiles in the samples were measured using D(He3,p)α nuclear reaction analysis. The D binding states in the defects were analysed using thermal desorption spectroscopy (TDS). In the case of irradiation at 290 K, the trapped D concentration monotonically increases with increasing Re concentration: in W-5%Re it is 17% higher than in pure W. On the contrary, for irradiation at 1350 K the trapped D concentration monotonically decreases with increasing Re concentration: in W-5%Re it is 50 times smaller than in pure W. In the case of annealing at 1350 K of the samples irradiated at 290 K, the Re presence yields only up to three times reduction of trapped D concentration compared with pure W. TDS shows that the nature of D trapping sites is different for the irradiations at 290 K and 1350 K. We attribute the reduced D trapping in W-Re alloys irradiated at 1350 K to the reduction of cavity growth caused by the presence of Re.