We study the asymptotic tail behaviour of the first passage time over a moving boundary for asymptotically $$\alpha $$ α -stable Lévy processes with $$\alpha <1$$ α < 1 . Our main result states that if the left tail of the Lévy measure is regularly varying with index $$- \alpha $$ - α , and the moving boundary is equal to $$1 - t^{\gamma }$$ 1 - t γ for some $$\gamma <1/\alpha $$ γ < 1 / α , then the probability that the process stays below the moving boundary has the same asymptotic polynomial order as in the case of a constant boundary. The same is true for the increasing boundary $$1 + t^{\gamma }$$ 1 + t γ with $$\gamma <1/\alpha $$ γ < 1 / α under the assumption of a regularly varying right tail with index $$-\alpha $$ - α .