Single-crystal elastic constants of a natural Fe-rich olivine (Fe0.94Mn0.06)2SiO4 were determined by Brillouin scattering to 12.1 GPa. The aggregate bulk modulus, shear modulus, and their pressure derivatives are KS0 = 136.3 (2) GPa, G0 = 51.2 (2) GPa, (∂KS/∂P)T0 = 4.9 (1), (∂G/∂P)0 = 1.8 (1), and (∂2G/∂P2)0 = -0.11 (1) GPa-1. The numbers in parentheses are 1σ uncertainties on the last digit. Our results demonstrate that both the bulk and the shear moduli of fayalite depend on the Fe/(Fe + Mg) ratio at both ambient and high pressures. The pressure derivative of the bulk modulus is very sensitive to the Fe content of olivine, while the pressure dependences of the shear modulus of Mg-rich and Fe-rich compositions are comparable. The longitudinal (C11, C22, C33) and the off-diagonal (C12, C13, C23) moduli of fayalite show a nearly linear dependence on pressure along the whole experimental pressure range. However, the shear constants C44 and C55 show a strongly nonlinear dependence on pressure starting at P > 5 GPa, more than 2 GPa above the extrapolated room temperature stability limit of fayalite. The behavior of the shear constants could be a precursor of the high-pressure amorphization of fayalite observed at 30 to 40 GPa. C44 is the modulus that shows the strongest tendency to soften with pressure, in disagreement with previous suggestions that C55 softening could trigger the structural transition of olivine to the high-pressure γ-Fe2SiO4. Softening of C44 and C55 is compatible with a diffusionless mechanism of the structural α- to γ-Fe2SiO4 transition.