Let A be a Banach algebra, let X and Y be Banach A-modules and take \({\mathfrak {B}}(X, Y)\) 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 \({\mathfrak {B}}(X, Y)\) . 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.
