Context. We investigated the molecular gas associated with 6.7 GHz methanol masers throughout the Galaxy using a J = 1−0 transition of the CO isotopologues. Aims. The methanol maser at 6.7 GHz is an ideal tracer for young high-mass star-forming cores. Based on molecular line emissions in the maser sources throughout the Galaxy, we can estimate their physical parameters and, thereby, investigate the forming conditions of the high-mass stars. Methods. Using the 13.7-m telescope at the Purple Mountain Observatory (PMO), we have obtained 12 CO and 13 CO (1−0) lines for 160 methanol masers sources from the first to the third Galactic quadrants. We made efforts to resolve the distance ambiguity by careful comparison with the radio continuum and HI 21 cm observations. We examined the statistical properties in three aspects: first, the variation throughout the Galaxy; second, the correlation between the different parameters; third, the difference between the maser sources and the infrared dark clouds. In addition, we have also carried out 13 CO mapping for 33 sources in our sample. Results. First, the maser sources show increased 13 CO line widths toward the Galactic center, suggesting that the molecular gas are more turbulent toward the Galactic center. This trend can be noticeably traced by the 13 CO line width. In comparison, the Galactic variation for the H 2 column density and the 12 CO excitation temperature are less significant. Second, the 12 CO excitation temperature shows a noticeable correlation with the H 2 column density. A possible explanation consistent with the collapse model is that the higher surface-density gas is more efficient to the stellar heating and/or has a higher formation rate of high-mass stars. Third, comparing the infrared dark clouds, the maser sources on average have significantly lower H 2 column densities, moderately higher temperatures, and similar line widths. Fourth, In the mapped regions around 33 masers, 51 13 CO cores have been revealed. Among them, only 17 coincide with the radio continuum emission (F cm > 6 mJy), while a larger fraction (30 cores) coincide with the infrared emissions. Only one maser source has no significant IR emission. The IR-bright and radio-bright sources exhibit significantly higher 12 CO excitation temperatures than the IR-faint and radio-faint sources, respectively. Conclusions. The 6.7 GHz masers show a moderately low ionization rate but have a common-existing stellar heating that generates the IR emissions. The relevant properties can be characterized by the 12 CO and 13 CO (1−0) emissions in several aspects as described above.