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In this work we demonstrate that wormholes can in principle be naturally created during the cosmological bounce without the need for the exotic matter or any kind of additional modifications of the gravitational sector, apart from the one enabling the cosmological bounce. This result is general and does not depend on the details of the modifications of gravitational equations needed to support the bounce. To study the possible existence of wormholes around the cosmological bounce we introduce general modifications of Einstein's field equations need to support the bouncing solutions. In this regime we show that it is possible to construct a cosmological wormhole solution supported by matter, radiation and vacuum energy, satisfying the Weak Energy Condition (WEC), which asymptotically approaches the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. However, at a specific cosmological time, which depends on the parameters of the bouncing cosmological model, the WEC describing the matter needed to support such wormholes is spontaneously violated. This means that such wormholes could potentially exist in large numbers during some period around the bounce, significantly changing the causal structure of space-time, and then vanish afterwards.