We present a Zeeman–Doppler imaging study of two weak-line T Tauri stars TAP 4 and TAP 40, based on the high-resolution spectropolarimetric observations with ESPaDOnS at the Canada–France–Hawaii Telescope in November 2013, in the framework of the Magnetic Topologies of Young Stars and Survival of close-in giant Exoplanets large programme. We apply two Zeeman–Doppler imaging codes to the Stokes I and V profiles to reconstruct their brightness and large-scale magnetic field images. The results given by the two imaging codes are in good agreement with each other. TAP 4 shows a large polar cool spot and several intermediate-latitude warm spots on its surface, whereas TAP 40 exhibits very weak variations in its Stokes I profiles, suggesting a mostly unspotted photosphere. We detect Zeeman signatures in the Stokes V profiles of both stars. The reconstructed magnetic maps reveal dominantly toroidal fields, which enclose about 60 per cent of the total magnetic energy for both TAP 4 and TAP 40. Both stars show prominent circular ring features of the azimuthal magnetic field. We derive a solar-like surface differential rotation on TAP 4 from the tomographic modelling. The brightness image of TAP 4 is used to predict the radial velocity (RV) jitters induced by its activity. After filtering out the activity jitter, the rms of its RVs is reduced from 1.7 to 0.2 km s−1, but we do not detect any periodic signals in the filtered RVs of TAP 4, implying that it is unlikely to host a close-in exoplanet more massive than ∼3.5MJup at 0.1 au. © 2023 The Author(s) Published by Oxford University Press on behalf of Royal Astronomical Society.