  The methanol molecule CH3OH has a complex microwave spectrum with a large
number of very strong lines. This spectrum includes purely rotational
transitions as well as transitions with contributions of the internal degree of
freedom associated with the hindered rotation of the OH group. The latter takes
place due to the tunneling of hydrogen through the potential barriers between
three equivalent potential minima. Such transitions are highly sensitive to
changes in the electron-to-proton mass ratio, mu = m_e/m_p, and have different
responses to mu-variations. The highest sensitivity is found for the mixed
rotation-tunneling transitions at low frequencies. Observing methanol lines
provides more stringent limits on the hypothetical variation of mu than ammonia
observation with the same velocity resolution. We show that the best quality
radio astronomical data on methanol maser lines constrain the variability of mu
in the Milky Way at the level of |Delta mu/mu| < 28x10^{-9} (1sigma) which is
in line with the previously obtained ammonia result, |Delta mu/mu| < 29x10^{-9}
(1\sigma). This estimate can be further improved if the rest frequencies of the
CH3OH microwave lines will be measured more accurately.
