This paper addresses the question whether normal forms of smooth reversible vector fields in R4 at an elliptic equilibrium possess a formal Hamiltonian structure. In the non-resonant case we establish a formal conjugacy between re-versible and Hamiltonian normal forms. In the case of non-semi-simple 1 : 1 resonance and p:q resonance with p+q >2 we establish a weaker form of equivalence, namely that of a formal orbital equivalence to a Hamiltonian normal formthat involves an additional time-reparametrization of orbits. Moreover, in case p+q >3 we show that no formal conjugacy to a Hamiltonian normal form exists.