Wasserstein barycenters do not preserve marginals in general. In this work, we first characterize sufficient and necessary conditions for the Wasserstein barycenter between two Gaussian distributions to preserve marginals, and provide necessary conditions in the case of more than two Gaussians. We then propose modified Wasserstein barycenters that preserve the marginals of the distributions, both for Gaussian distributions and for mixtures of Gaussian distributions. In the case of Gaussian distributions, the marginal-preserving modified Wasserstein barycenters can be analytically computed, while for Gaussian mixtures, computing the marginal-preserving barycenter consists in a postprocessing of the Gaussian mixture Wasserstein barycenter. In both cases, we provide numerical simulations illustrating the difference between Wasserstein barycenters and modified marginal-preserving Wasserstein barycenters.