An associative ring \(R\) with identity is called left morphic if for every element \(a\in R\) there exists \(b\in R\) such that \(l_R(a)=Rb\) and \(l_R(b)=Ra\), where \(l_R(a)\) denotes the left annihilator of \(a\) in \(R\). The ring \(R\) is said to be strongly left morphic if every matrix ring \(M_n(R)\) is left morphic [\textit{W. K. Nicholson} and \textit{E. Sánchez Campos}, J. Algebra 271, No. 1, 391-406 (2004; Zbl 1071.16006)]. In this paper, the authors discuss the morphic problem for group rings. They prove that if the group ring \(RG\) is left morphic, then \(R\) is left morphic and \(G\) is a locally finite group. Conversely, if \(G\) is a locally finite group and \(RH\) is left morphic for every finite subgroup \(H\) of \(G\), then \(RG\) is also left morphic. The case when \(R=Z_{p^r}\) and \(G\) is finite is considered, too. In great details these problems are considered when \(G\) is a finite Abelian group. It is achieved progress on the cases when \(R\) is semisimple or \(R=\mathbb{Z}_n\). Finally, the attention is concentrated on the group ring \(\mathbb{Z}_mD_n\), where \(D_n\) is the dihedral group.