This question cannot be answered within ZFC, and the answer depends somewhat on the answerer's philosophical position. It's somewhat related to the question "Are proper classes objects?" on MathOverflow, and the answer someone gives to this question is going to be related to the answer she gives to that question. Unfortunately, as you will see by looking there, both the answer "Yes, proper classes are objects" and the answer "No, proper classes are not objects" have proponents. Similarly, some people would answer this question "Yes", although most set theorists would answer it "No".
For those who accept there is a well-defined, objective meaning to the term "all sets", the answer to this question is "No". Under this viewpoint, the collection of "all sets" is a class model of ZFC which cannot be a set in any other (transitive) model of ZFC, because the collection of "all sets" is itself not a set by Russell's paradox. This is by far the most common position on your question among set theorists.
To argue that every class model of ZFC is a set in a larger (transitive) model of ZFC seems to require giving up the idea that "all sets" has an objective meaning apart from a model of ZFC. In that case, it is consistent that every class model of ZFC is embeddable as a set in a different model of ZFC, and in fact it's consistent that every class model of ZFC is countable in another model of ZFC. By "consistent" I mean that this situation is relatively consistent with ZFC itself, as shown by Gitman and Hamkins in their recent paper "A natural model of the multiverse axioms". However, few set theorists would agree that the multiverse axioms are true.