You may look at the paper ``Forcing with non-wellfounded models''. Australas. J. Log. 5 (2007), 20–57, by Paul Corazza:
Here is the abstract of the paper:
We develop the machinery for performing forcing over an arbitrary (possibly non-well-founded) model of set theory. For consistency results, this machinery is unnecessary since such results can always be legitimately obtained by assuming that the ground model is (countable) transitive. However, for establishing properties of a given (possibly non-well-founded) model, the fully developed machinery of forcing as a means to produce new related models can be useful. We develop forcing through iterated forcing, paralleling standard steps of presentation in [K. Kunen, Set theory, North-Holland, Amsterdam, 1980] and [T. J. Jech, in Handbook of Boolean algebras, Vol. 3, 1197–1211, North-Holland, Amsterdam, 1989].''