This study is purposed to elaborate the problem of energy and momentum distributions of the Hawking wormholes in general theory of relativity and teleparallel gravity. In this connection, the energy and momentum for the Hawking wormhole space-time are evaluated using Einstein, Bergmann-Thomson, Landau-Lifshitz and Moller energy-momentum prescriptions in these approximations. We obtained that some of these energy-momentum distributions of Hawking wormhole in the form of general relativity (GR) and teleparallel gravity (TG) give the same results. Also, we obtained that in the teleparallel gravity, Moller energy distributions of the Hawking wormholes give the same results as Einstein energy distributions. However, Bergmann-Thomson and Landau-Lifshitz energy-momentum of Hawking wormholes are the different from each others.