The problem of moment matching with preservation of port Hamiltonian structure is tackled. Based on the time-domain approach to linear moment matching, we characterize the (subset of) port Hamiltonian models from the set of parameterized models that match the moments of a given port Hamiltonian system, at a set of finite points. We also discuss the problem of finding port Hamiltonian reduced order models that match the Markov parameters of a given port Hamiltonian system.