PSR J1357—6429 is a young radio pulsar detected in X-rays and γ-rays. We performed radio-interferometric and high spatial resolution near-infrared observations of the pulsar. Based on the obtained radio data, we estimated the most accurate pulsar position, RA = 13:57:02.525(14) and Dec = 64:29:29.89(15). Using the new and archival radio data, we obtained the 90 per cent pulsar proper motion upper limit μ < 106 mas yr-1, which corresponds to the pulsar transverse velocity vtr < 1300 km s-1. The near-infrared imaging revealed a point source which we propose as a pulsar near-infrared counterpart candidate. It is confidently detected in the J and Ks bands, with J = 23.51±0.24 and Ks = 21.82±0.25. The respective dereddened near-infrared fluxes are compatible with the long-wavelength extrapolation of the pulsar X-ray spectrum. If the candidate is the true counterpart, by this property PSR J1357—6429 would be similar to the nearby middle-age pulsar PSR B0656+14. In this case, both pulsars demonstrate an unusually high near-infrared efficiency relative to the X-ray efficiency as compared to other pulsars detected in both ranges.