We propose a new approximate value iteration method, namely near-value iteration (NVI), to solve continuous-state optimal stopping problems under partial observation, which in general cannot be solved analytically and also pose a great challenge to numerical solutions. NVI is motivated by the expression of the value function as the supremum over an uncountable set of linear functions in the belief state. After a smart manipulation of the operations in the updating equation for the value function, we reduce the set to only two functions at every time step, so as to achieve significant computational savings. NVI yields a value function approximation bounded by the tightest lower and upper bounds that can be achieved by existing algorithms in the same class, so the NVI approximation is closer to the true value function than at least one of these bounds. We demonstrate the effectiveness of our approach on an example of pricing American options under stochastic volatility.