Phase equilibria for mixtures of a polymer and a supercritical solvent are investigated by means of expanded Gibbs ensemble simulations. Both lower and upper critical solution temperature (LCST and UCST) phenomena are observed for such systems. A closed-loop phase diagram is observed for systems with no specific interactions. The results of our simulations for Lennard-Jones polymer-solvent mixtures are in qualitative agreement with experimental data for polymers in supercritical solvents.