A finite horizon optimal stopping problem for an infinite dimensional diffusion X is analyzed by means of variational techniques. The diffusion is driven by a SDE on a Hilbert space with a non-linear diffusion coefficient and a generic unbounded operator A in the drift term. When the gain function is time-dependent and fulfils mild regularity assumptions, the value function of the optimal stopping problem is shown to solve an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. Once the coefficient is specified, the solution of the variational problem is found in a suitable Banach space fully characterized in terms of a Gaussian measure . This work provides the infinite-dimensional counterpart, in the spirit of Bensoussan and Lions (Application of variational inequalities in stochastic control, 1982), of well-known results on optimal stopping theory and variational inequalities in . These results may be useful in several fields, as in mathematical finance when pricing American options in the HJM model.