This work addresses the problem of pricing American basket options in a multivariate setting, which includes among others, the Bachelier and Black–Scholes models. In high dimensions, nonlinear PDE methods for solving the problem become prohibitively costly due to the curse of dimensionality. Instead, this work proposes to use a stopping rule that depends on the dynamics of a low-dimensional Markovian projection of the given basket of assets. From a numerical analysis point of view, we split the given non-smooth high-dimensional problem into two subproblems, namely one dealing with a smooth high-dimensionality integration in the parameter space and the other dealing with a low-dimensional, non-smooth optimal stopping problem in the projected state space. Assuming that we know the density of the forward process and using the Laplace approximation, we first efficiently evaluate the diffusion coefficient corresponding to the low-dimensional Markovian projection of the basket. Then, we approximate the optimal early exercise boundary of the option by solving an HJB PDE in the projected, low-dimensional space. The resulting near-optimal early exercise boundary is used to produce an exercise strategy for the high-dimensional option, thereby providing a lower bound for the price of the American basket option. A corresponding upper bound is also provided. These bounds allow one to assess the accuracy of the proposed pricing method. Indeed, our approximate early exercise strategy provides a straightforward lower bound for the American basket option price. Following a duality argument due to Rogers, we derive a corresponding upper bound solving only the low-dimensional optimal control problem. Numerically, we show the feasibility of the method using baskets with dimensions up to 50. In these examples, the resulting option price relative errors are only of the order of few percent.