The La Liga giants are understood to have made an improved bid for the 22-year-old worth £145.9m plus £8.6m in guaranteed add-ons, after having an opening offer of £137m rejected on Wednesday.
Mbappe has less than 12 months remaining on his contract and has told the Parisian club of his wishes to seal a dream move to Los Blancos, whether it is this summer or next year.
According to Globo journalist Alexandre Lozetti, PSG are preparing an offer to sign Jesus as soon as Mbappe's move to Spain in confirmed.
Pep Guardiola lavished praise on the 24-year-old for his impressive performance in last weekend's 5-0 win over Norwich City and sees him as part of his plans going forward.
PSG are also interested in Everton's Richarlison, though Rafael Benitez is adamant that the Toffees have no intention of selling the Brazilian.