We show that instanton bundles of rank r <= 2n - 1, defined as the cohomology of certain linear monads, on an n-dimensional projective variety with cyclic Picard group are semistable in the sense of Mumford-Takemoto. Furthermore, we show that rank r <= n linear bundles with nonzero first Chern class over such varieties are stable. We also show that these bounds are sharp.361288298