Galaxy clusters observed through the thermal Sunyaev-Zeldovich ( tSZ ) effect are a recent cosmological probe . The precision on the cosmological constraints is affected mainly by the current knowledge of cluster physics , which enters the analysis through the scaling relations . Here we aim to study one of the most important sources of systematic uncertainties , the mass bias , b . We have analysed the effects of a mass-redshift dependence , adopting a power-law parametrisation . We applied this parametrisation to the combination of tSZ number counts and power spectrum , finding a hint of redshift dependence that leads to a decreasing value of the mass bias for higher redshift . We tested the robustness of our results for different mass bias calibrations and a discrete redshift dependence . We find our results to be dependent on the clusters sample that we are considering , in particular obtaining an inverse ( decreasing ) redshift dependence when neglecting z < 0.2 clusters . We analysed the effects of this parametrisation on the combination of cosmic microwave background ( CMB ) primary anisotropies and tSZ galaxy clusters . We find a preferred constant value of mass bias , having ( 1 - b ) = 0.62 \pm 0.05 . The corresponding value of b is too high with respect to weak lensing and numerical simulations estimations . Therefore we conclude that this mass-redshift parametrisation does not help in solving the remaining discrepancy between CMB and tSZ clusters observations .