Context. Colliding winds in massive binaries are able to accelerate particles up to relativistic speeds as the result of the interaction between the winds of the different stellar components. HD 167971 exhibits this phenomenon which makes it a strong radio source. Aims. We aim at characterizing the morphology of the radio emission and its dependence on the orbital motion, traced independently by near-infrared (NIR) interferometry of both the spectroscopic binary and the tertiary component comprising HD 167971. Methods. We analyze 2006 and 2016 very long baseline interferometric data at C and X bands. We complement our analysis with a geometrical model of the wind-wind collision region and an astrometric description of the system. Results. We confirm that the detected nonthermal radio emission is associated with the wind-wind collision region of the spectroscopic binary and the tertiary component in HD 167971. The wind-wind collision region changes orientation in agreement with the orbital motion of the tertiary around the spectroscopic binary. The total intensity also changes between the two observing epochs in a way that is inversely proportional to the separation between the two components, with a negative-steep spectral index typical of an optically thin synchrotron emission possibly steepened by an inverse Compton cooling effect. The wind-wind collision bow-shock shape and its position with respect to the stars indicates that the wind momentum from the spectroscopic binary is stronger than that of the tertiary. Finally, the astrometric solution derived for the stellar system and the wind-wind collision region is consistent with independent Gaia data.