The possible existence of four-neutron resonances close to the physical energy region is explored. Faddeev-Yakubovsky equations have been solved in configuration space using realistic nucleon-nucleon interaction models. Complex scaling and analytical continuation in the coupling constant methods were used to follow the resonance pole trajectories, which emerge out of artificially bound tetraneutron states. The final pole positions for four-neutron states lie in the third energy quadrant with negative real energy parts and should thus not be physically observable.