The concept of Hopfian modules has been extensively studied in the literature. In this paper, we introduce and study the notion of δ-weakly Hopfian modules. The class of δ-weakly Hopfian modules lies properly between the class of Hopfian modules and the class of weakly Hopfian modules. It is shown that over a ring R, every quasiprojective R-module is δ-weakly Hopfian iff δ(R) = J (R). We prove that any weak duo module, with zero radical is δ-weakly Hopfian. Let M be a module such that M satisfies ascending chain condition on δ-small submodules. Then it is shown that M is δ-weakly Hopfian. Some other properties of δ-weakly Hopfian modules are also obtained with examples.
