Studies of young stellar objects (YSOs) in the Galaxy have found that a significant fraction exhibits photometric variability. However, no systematic investigation has been conducted on the variability of extragalactic YSOs. Here we present the first variability study of massive YSOs in an ~1.5 square degree region of the Large Magellanic Cloud (LMC). The aim is to investigate whether the different environmental conditions in the metal-poor LMC (~0.4-0.5Z_{sun}_) have an impact on the variability characteristics. Multi-epoch near-infrared (NIR) photometry was obtained from the VISTA Survey of the Magellanic Clouds (VMC) and our own monitoring campaign using the VISTA telescope. By applying a reduced {Chi}-square-analysis, stellar variability was identified. We found 3062 candidate variable stars from a population of 362425 stars detected. Based on several Spitzer studies, we compiled a sample of high-reliability massive YSOs: a total of 173 massive YSOs have NIR counterparts (down to Ks~18.5mag) in the VMC catalogue, of which 39 display significant (>3{sigma}) variability. They have been classified as eruptive, fader, dipper, short-term variable, and long-period variable YSOs based mostly on the appearance of their Ks-band light curves. The majority of YSOs are aperiodic; only five YSOs exhibit periodic light curves. The observed amplitudes are comparable or smaller than those for Galactic YSOs (only two Magellanic YSOs exhibit {Delta}Ks>1mag), not what would have been expected from the typically larger mass accretion rates observed in the Magellanic Clouds.