Many algorithms for query rewriting using views have been proposed. Most of them are used for rewriting conjunctive queries with conjunctive views only. Only one inverse rule-based rewriting algorithm has been presented to deal with query rewriting with disjunctive views, in which a set of disjunctive inverse rules are created and used to generate the query rewritings. However, there has no been bucket-based algorithm for this issue. In this paper we apply the ideas of the previous bucket-based algorithms to deal with query rewriting using views in the presence of disjunctions in view definitions. We create a set of buckets over either a conjunctive view or a disjunctive view. Specially, when a bucket is over a disjunctive view, we present a method to remove the tuples that are in the view but not required by a given query. The rewritings obtained by our algorithm are contained in the original query.