We consider the problem of deciding whether a query can be rewritten by a nondeterministic view. It is known that rewriting is decidable if views are given by single-valued non-copying devices such as compositions of single-valued extended linear top-down tree transducers with regular look-ahead, and queries are given by deterministic MSO tree transducers. In this paper, we extend the result to the case that views are given by nondeterministic devices that are not always single-valued. We define two variants of rewriting: universal preservation and existential preservation, and discuss the decidability of them.