Providing up-to-date input to users' applications is an important data management problem for a distributed computing environment, where each data storage location and intermediate node may have specific data available, storage limitations, and communication links available. Sites in the network request data items and each request has an associated deadline and priority. This work concentrates on solving a basic version of the data staging problem in which all parameter values for the communication system and the data request information represent the best known information collected so far and stay fixed throughout the scheduling process. The network is assumed to be oversubscribed and not all requests for data items can be satisfied. A mathematical model for the basic data staging problem is given. Then, three multiple-source shortest-path algorithm based heuristics for finding a near-optimal schedule of the communication steps for staging the data are presented. Each heuristic can be used with each of four cost criteria developed. Thus, twelve implementations are examined. In addition, two different weightings for the relative importance of different priority levels are considered. The performance of the proposed heuristics is evaluated and compared by simulations. The proposed heuristics are shown to perform well with respect to upper and lower bounds. Furthermore, the heuristics and a complex cost criterion allow more highest priority messages to be received than a simple-cost-based heuristic that schedules all highest priority messages first.