It has been argued that the crystallization of the magma ocean (MO) after the Moon-forming impact led to the formation of a basal magma ocean (BMO). We search which primordial conditions of pressure, temperature and chemical composition could be compatible with such scenario, based on thermodynamical constraints. The major requirement is an early formation of a viscous layer (VL) of mantle material (i.e. bridgmanite (Bg)) at mid lower-mantle depth, which could insulate thermally and chemically the BMO from the rest of the mantle. To produce such VL, Bg grains should be: (i) neutrally buoyant at mid lower-mantle depths, (ii) sufficiently abundant to produce an efficient insulating layer, and (iii) aggregated to the boundary layer from above and below. The first and the second require a large amount of MO crystallization, up to more than 45%, even in the most favorable case of all Fe partitioning into the melt. The latter is very questionable because the Bg grains have a very small settling velocity. We also investigate different scenarios of MO crystallization to provide constraints on the resulting core temperature. Starting from a fully molten Earth, a temperature as high as ∼4725K could be found at the core–mantle boundary (CMB), if the Bg grains settle early atop the CMB. Such a basal layer of Bg can efficiently decouple from each other the cooling rates of the core and the mantle above the VL. If the settling velocity of Bg grains is too low and/or the MO is too turbulent, such basal VL may not form. In this case, the CMB temperature after MO solidification should stabilize at ∼4350K. At this temperature, enough Bg grains are crystallized to make the mushy mantle viscous at any mantle depth.