Theoretical simulations and observations at different angular resolutions have shown that magnetic fields have a central role in massive star formation. Like in low-mass star formation, the magnetic field in massive young stellar objects can either be oriented along the outflow axis or randomly. Measuring the magnetic field at milliarcsecond resolution (10-100au) around a substantial number of massive young stellar objects permits determining with a high statistical significance whether the direction of the magnetic field is correlated with the orientation of the outflow axis or not. In late 2012, we started a large VLBI campaign with the European VLBI Network to measure the linearly and circularly polarized emission of 6.7GHz CH_3_OH masers around a sample of massive star-forming regions. This paper focuses on the first seven observed sources, G24.78+0.08, G25.65+1.05, G29.86-0.04, G35.03+0.35, G37.43+1.51, G174.20-0.08, and G213.70-12.6. For all these sources, molecular outflows have been detected in the past. We detected a total of 176 CH_3_OH masing cloudlets toward the seven massive star-forming regions, 19% of which show linearly polarized emission. The CH_3_OH masers around the massive young stellar object MM1 in G174.20-0.08 show neither linearly nor circularly polarized emission. The linear polarization vectors are well ordered in all the other massive young stellar objects. We measured significant Zeeman splitting toward both A1 and A2 in G24.78+0.08, and toward G29.86-0.04 and G213.70-12.6. By considering all the 19 massive young stellar objects reported in the literature for which both the orientation of the magnetic field at milliarcsecond resolution and the orientation of outflow axes are known, we find evidence that the magnetic field (on scales 10-100au) is preferentially oriented along the outflow axes.