Covariance intersection (CI) is a method used for consistent track fusion with unknown correlations. The well-known generalization of CI to probability density functions is known as Chernoff fusion. In this paper, we propose an approximate approach for the Chernoff fusion of Gaussian mixtures based on a sigma-point approximation of the underlying densities. The resulting general density fusion rule yields a closed form cost function and an analytical fused density for Gaussian mixtures. The proposed method is applied to a simple but illustrative density fusion problem and compared to exact numerical Chernoff fusion.