The spatiotemporal correlations of the local stress tensor in supercooled liquids are studied both theoretically and by molecular dynamics simulations of a two-dimensional (2D) polydisperse Lennard-Jones system. Asymptotically exact theoretical equations defining the dynamical structure factor and all components of the stress correlation tensor for low wave-vector q are presented in terms of the generalized ( q-dependent) shear and longitudinal relaxation moduli, G( q, t) and K( q, t). We developed a rigorous approach (valid for low q) to calculate K( q, t) in terms of certain bulk correlation functions (for q = 0), the static structure factor S( q), and thermal conductivity κ. The proposed approach takes into account both the thermostatting effect and the effect of polydispersity. The theoretical results for the ( q, t)-dependent stress correlation functions are compared with our simulation data, and an excellent agreement is found for [Formula: see text] (with [Formula: see text] being the mean particle diameter) both above and below the glass transition without any fitting parameters. Our data are consistent with recently predicted (both theoretically and by simulations) long-range correlations of the shear stress quenched in heterogeneous glassy structures.