Let \(R\) be a commutative ring. Recall that a submodule \(N\) of an \(R\)-module \(M\) is said to be pure if \(IN = N \cap IM\), for every ideal \(I\) of \(R\). In [``Strong comultiplication modules'', CMU. J. Nat. Sci. 8, No. 1, 105--114 (2009)], \textit{H. Ansari-Toroghy} and the author introduced a ``dual'' notion of a pure submodule, called copure submodule, and investigated the basic properties of this class of modules. More precisely, a submodule \(N\) of an \(R\)-module \(M\) is said to be copure if (\(N :_M I) = N + (0 :_M I)\), for every ideal \(I\) of \(R\). On the other hand, following the concept of 2-absorbing ideals (after A. Badawi), the author of the present paper has recently introduced the concept of a 2-absorbing pure submodule of an \(R\)-module \(M\), as a generalization of a pure submodule. A submodule \(N\) of an \(R\)-module \(M\) is said to be a 2-absorbing pure submodule of \(M\) if \(IJN = IN \cap JN \cap IJM\), for every pair of ideals \(I\) and \(J\) of \(R\); an ideal \(I\) of \(R\) is said to be a 2-absorbing pure ideal of \(R\) if \(I\) is a 2-absorbing pure submodule of \(R\). The main purpose of this paper is to introduce the concept of 2-absorbing copure submodules of an \(R\)-module \(M\), as a natural generalization of copure submodules, and to investigate the basic properties of this class of submodules and of the class of copure submodules.