Massive young stellar objects (YSOs), like low-mass YSOs, are thought to be surrounded by optically thick envelopes and/or discs and are observed to have associated regions that produce polarized light at near-infrared wavelengths. These polarized regions are thought to be lower-density outflows along the polar axes of the YSO envelopes. Using the 0.2 arcsec spatial resolution of the Near-Infrared Camera and Multi-Object Spectrometer on the Hubble Space Telescope we are examining the structure of the envelopes and outflow regions of massive YSOs in star-forming regions within a few kpc of the Sun. Here we report on 2 micron polarimetry of Mon R2-IRS3, S140-IRS1, and AFGL 2591. All three sources contain YSOs with highly-polarized monopolar outflows, with Mon R2-IRS3 containing at least two YSOs in a small cluster. The central stars of all four YSOs are also polarized, with position angles perpendicular to the directions of the outflows. We infer that this polarization is due to scattering and absorption by aligned grains. We have modelled our observations of S140-IRS1 and AFGL 2591 as light scattered and absorbed both by spherical grains and by elongated grains that are aligned by magnetic fields. Models that best reproduce the observations have a substantial toroidal component to the magnetic field in the equatorial plane. Moreover, the toroidal magnetic field in the model that best fits AFGL 2591 extends a large fraction of the height of the model cavity, which is 10^5 au. We conclude that the massive YSOs in this study all show evidence of the presence of a substantial toroidal magnetic field. ; Comment: 20 pages, 14 figures. To be published in MNRAS