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Let R be an associative ring with identity and M be a left R-module. In this paper, we define modules that have the property (δ- CE) ((δ-CEE)), these are modules that have a δ-supplement (ample δ- supplements) in every cofinite extension which are generalized version of modules that have the properties (CE) and (CEE) introduced in [6] and so a generalization of Zöschinger’s modules with the properties (E) and (EE) given in [23]. We investigate various properties of these modules along with examples. In particular we prove these: (1) a module M has the property (δ-CEE) if and only if every submodule of M has the property (δ-CE); (2) direct summands of a module that has the property (δ-CE) also have the property (δ-CE); (3) each factor module of a module that has the property (δ-CE) also has the property (δ-CE) under a special condition; (4) every module with composition series has the property (δ- CE); (5) over …