Aims. We investigate whether viscous cold dark matter (vCDM) in a Λ-dominated Friedmann-Lemaître-Robertson-Walker (FLRW) universe can alleviate the Hubble tension, while satisfying thermodynamic constraints. Here, we examine both flat and curved geometries. Methods. We modeled a vCDM cosmology with a bulk viscosity ζ = ζ0 (Ωvc/Ωvc0)m, where m determines the viscosity evolution and Ωvc is the density parameter of vCDM. We explored two particular scenarios: (i) constant viscosity ζ = ζ0 (m = 0) and (ii) variable viscosity ζ = ζ(Ωvc) (m free). Using Bayesian inference, we constrained these models with the latest datasets: the Pantheon+ Type Ia supernova sample (with a SH0ES calibration, PPS, and without, PP), Hubble parameter measurements, H(z), from cosmic chronometers and baryon acoustic oscillations (BAO) as independent datasets, including DESI DR2, and a Gaussian prior on H0 from the SH0ES measurement, H0 = 73.04 ± 1.04 km s−1 Mpc−1 (R22 prior). We compared the models via Akaike, Bayesian, and deviance information criteria (AIC, BIC, and DIC), and with Bayesian evidence. Results. Our results indicate that the Hubble tension persists, although it shows partial alleviation (∼1σ tension) in all investigated scenarios when local measurements are included. For the flat m = 0 case, the joint analysis yields H0 = 71.05+0.62−0.60 H 0 = 71 . 05 − 0.60 + 0.62 km s−1 Mpc−1. Curved models initially favor ΩK0 > 0 (at more than 2σ), but this preference shifts toward flatness once the PPS + R22 prior is included. Notably, the current viscosity is constrained to ζ0 ∼ 106 Pa s in all scenarios, in agreement with the thermodynamic requirements. Although the model selection via BIC and Bayesian evidence favors ΛCDM, the AIC and DIC show mild support for viscous models in some datasets. Conclusions. Bulk viscous models moderately improve the fits, but they can neither resolve the Hubble tension nor outperform the ΛCDM model. To achieve more robust constraints, future analyses should incorporate CMB observations, which are expected to break parameter degeneracies involving m and ζ∼0 ζ ∼ 0 .