Moti Gitik had prove that there exists a model of ZF in which every aleph bigger than $\aleph_0$ is of $\aleph_0$ cofinality, call such models as Moti Gitik models.
Supposing that there exists an $\omega$-model of ZF. Would it follow that there can exist an $\omega$-Moti Gitik model.
An $\omega$-model is a model in which all naturals are the standard naturals as externally seen.
Gitik's model is obtained by forcing methods, these do not change the ordinals, so in particular preserve things like $\omega$-model.
However, the requirements for Gitik's model are currently not known to be weaker than "a proper class of strongly compact cardinals", which exceed "there is an $\omega$-model" by a lot. And while it is conjectured that a similar result can be obtained with a lot less than even a single strongly compact, we do know that it implies there are many Woodin cardinals in an inner model, so you cannot bring this result anywhere near "there is an $\omega$-model".