  Methanol is a key species in astrochemistry since it is the most abundant
organic molecule in the ISM and is thought to be the mother molecule of many
complex organic species. Estimating the deuteration of methanol around young
protostars is of crucial importance because it highly depends on its formation
mechanisms and the physical conditions during its moment of formation. We
analyse dozens of transitions from deuterated methanol isotopologues coming
from various existing observational datasets from the IRAM-PdBI and ALMA sub-mm
interferometers to estimate the methanol deuteration surrounding three low-mass
protostars on Solar System scales. A population diagram analysis allows us to
derive a [CH$_2$DOH]/[CH$_3$OH] abundance ratio of 3-6 % and a
[CH$_3$OD]/[CH$_3$OH] ratio of 0.4-1.6 % in the warm inner protostellar
regions. These values are ten times lower than those derived with previous
single-dish observations towards these sources but they are 10-100 times higher
than the methanol deuteration measured in massive hot cores. Dust temperature
maps obtained from Herschel and Planck observations show that massive hot cores
are located in warmer molecular clouds than low-mass sources, with temperature
differences of $\sim$10 K. Comparison with the predictions of the gas-grain
astrochemical model GRAINOBLE shows that such a temperature difference is
sufficient to explain the different deuteration observed in low- to high-mass
sources, suggesting that the physical conditions of the molecular cloud at the
origin of the protostars mostly govern the present observed deuteration of
methanol. The methanol deuteration measured in this work is higher by a factor
of 5 than the upper limit in methanol deuteration estimated in comet Hale-Bopp,
implying that an important reprocessing of the organic material would have
occurred in the solar nebula during the formation of the Solar System.
