Faltings' annihilator theorem is an important result in local cohomology theory. Recently, Doustimehr and Naghipour generalized the Falitings' annihilator theorem. They proved that if R is a homomorphic image of a Gorenstein ring, then f b a (M )n = λ b a (M )n, where f b a (M )n := inf{i ∈ N | dim Supp(b t H i a (M )) ≥ n for all t ∈ N} and λ b a (M )n := inf{λ bRp aRp (Mp) | p ∈ Spec R with dim R/p ≥ n}. In this paper, we study the relation between f b a (M )n and λ b a (M )n, and prove that if R is an almost Cohen-Macaulay ring, then f b a (M )n ≥ λ b a (M )n − cmd R. Using this result, we prove that if R is a homomorphic image of a Cohen-Macaulay ring, then f b a (M )n = λ b a (M )n.