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.
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