ABSTRACT Utilizing the well-established radial Tully–Fisher (RTF) relation observed in a ‘large’ (843) sample of local galaxies, we report the maximum allowed variance in the Hubble parameter, $H_{0}$. We estimate the total intrinsic scatter in the magnitude of the RTF relation(s) implementing a cosmological model-independent cosmographic expansion. We find that the maximum allowed local ‘radial’ variation in our baseline analysis, using four RTF relations in the galaxy sample is $\Delta H_0/H_0\lesssim 3\ \hbox{per cent}$ at a 95 per cent C.L. significance, which is implied form a constraint of $\Delta H_0/H_0= 0.54^{+1.32}_{-1.37}\ \hbox{per cent}$ estimated at $D_{\rm L}\sim 10\, [{\rm Mpc}]$. Using only one ‘best-constrained’ radial bin, we report a conservative 95 per cent C.L. limit of $\Delta H_0/H_0\lesssim 4\ \hbox{per cent}$. Through our estimate of maximum variation, we propose a novel method to validate several late-time/local modifications put forth to alleviate the $H_0$ tension. We find that within the range of the current galaxy sample redshift distribution $10 \, [{\rm Mpc}] \le D_{\rm L}\le 140\, [{\rm Mpc}]$, it is highly unlikely to obtain a variation of $\Delta H_0/H_0\sim 9\ \hbox{per cent}$, necessary to alleviate the $H_0$-tension. However, we also elaborate on the possible alternative inferences when the innermost radial bin is included in the analysis. Alongside the primary analysis of fitting the individual RTF relations independently, we propose and perform a joint analysis of the RTF relations useful to create a pseudo-standardizable sample of galaxies. We also test for the spatial variation of $H_0$, finding that the current samples’ galaxies distributed only in the Southern hemisphere support the null hypothesis of isotropy within the allowed noise levels.