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Saddle dynamics is a time continuous dynamics to efficiently compute the any-index saddle points and construct the solution landscape. In practice, the saddle dynamics needs to be discretized for numerical computations, while the corresponding numerical analyses are rarely studied in the literature, especially for the high-index cases. In this paper we propose the convergence analysis of discrete high-index saddle dynamics. To be specific, we prove the local linear convergence rates of numerical schemes of high-index saddle dynamics, which indicates that the local curvature in the neighborhood of the saddle point and the accuracy of computing the eigenfunctions are main factors that affect the convergence of discrete saddle dynamics. The proved results serve as compensations for the convergence analysis of high-index saddle dynamics and are substantiated by numerical experiments. [ABSTRACT FROM AUTHOR]