In compressive sensing, one important parameter that characterizes the various greedy recovery algorithms is the iteration bound which provides the maximum number of iterations by which the algorithm is guaranteed to converge. In this letter, we present a new iteration bound for CoSaMP by certain mathematical manipulations including formulation of appropriate sufficient conditions that ensure passage of a chosen support through the two selection stages of CoSaMP, Augment and Update. Subsequently, we extend the treatment to the subspace pursuit (SP) algorithm. The proposed iteration bounds for both CoSaMP and SP algorithms are seen to be improvements over their existing counterparts, revealing that both CoSaMP and SP algorithms converge in fewer iterations than suggested by results available in literature.