Let A be a Banach algebra, let X and Y be Banach A-modules and take B(X, Y ) as the space of all bounded linear maps from X into Y . The pair (X, Y )A is said to be an AMHNMH if almost module homomorphisms are near module homomorphisms in the norm topology on B(X, Y ). We study AMHNMH pairs and give some examples of them. As a result, we prove that if A is amenable, X is a commutative Banach A-module and Y is a dual commutative Banach A-module, then the pair (X, Y )A is AMHNMH. As special cases, we also investigate the nearness of almost multipliers and almost derivations to multipliers and derivations respectively.
