On page 2 in "The Hyperuniverse Project and Maximality" is written:
The models $\mathcal{M}$ of MK are of the form $\langle M, \in, \mathcal{C} \rangle$, where $M$ is transitive model of ZFC, $\mathcal{C}$ the family of classes of $\mathcal{M}$ (i.e. every element of $\mathcal{C}$ is a subset of $M$) and $\in$ is the standard $\in$ relation.
Why can we assume that $M$ is transitive? If we knew, that every model of MK is well-founded, we could use the Mostowski collapse lemma. But is this the case?
Remark: In the book MK is formulated in a two-sorted version (have a look at the link).
As Caicedo pointed out in the comments by "models" only "standard models" are meant. The argument that there are illfounded models of MK is exactly the same as for ZFC (e.g. here). If we assume the axiom of foundation at the background level such an illfounded model is necessarily not isomorphic to a transitive model. If we don't have foundation in the metatheory this is not always the case (see this question).