The pressure coefficients of band gaps of silicon are calculated with use of a first-principles quasiparticle approach. For comparison, the same quantities are also computed within the Hohenberg-Kohn-Sham local-density approximation (LDA). In both cases, the calculations were performed for volumes corresponding to pressures between 0 and 150 kbar using ab initio norm-conserving pseudopotentials. The electron self-energy in the quasiparticle calculation is obtained with use of a first-order expansion in the dressed Green’s function G and the screened Coulomb interaction Wwith local fields. The quasiparticle results show a linear dependence of changes in the band gaps on lattice constant, while the dependence on pressure deviates from linearity at high pressure. The LDA band-gap pressure coefficients, obtained using three commonly used exchange-correlation potential forms, are shown to be very close to the quasiparticle values for the whole range of crystal volumes we considered. At some of the volumes considered, the LDA band gaps are negative. Both sets of pressure-coefficient results are in very good agreement with available experimental values for Si.