In this paper, we present a Gaussian approximation to the nonlinear filtering problem, namely the quasi-Gaussian Kalman filter. Starting with the recursive Bayes filter, we invoke the Gaussian approximation to reduce the filtering problem into an optimal Kalman recursion. We use the moment evolution equations for stochastic dynamic equations to evaluate the prediction terms in the Kalman recursions. We propose two methods, one based on stochastic linearization and the other based on a direct evaluation of the innovations terms, to perform the measurement update in the Kalman recursion. We test our filter on a simple two dimensional example, where the nonlinearity of the system dynamics and the measurement equations can be varied, and compare its performance to that of an extended Kalman filter.