The p-center problem has been the subject of interest in the operational research for a long time. It has been well-known since the middle of the previous century. During the last decade, an extension of the problem, known as the p-next center problem, has been defined in order to handle unexpected incidents that can disable the centers. There are only a few papers and algorithms that address the aforementioned problem and therefore we introduce a new algorithm for solving the p-next center problem based on the Variable Neighborhood Search Method. The proposed algorithm was tested on a set of test instances already known in the literature, and the results show that it returns an optimal or at least near-optimal solution to the problem in a reasonable amount of time. Compared to existing algorithms, it has been shown that the proposed algorithm finds the best known or better solutions.