Summary: 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 (\(\delta\)-\(CE\)) ((\(\delta\)-\(CEE\))), these are modules that have a \(\delta\)-supplement (ample \(\delta\)-supplements) in every cofinite extension which are generalized version of modules that have the properties (\(CE\)) and (\(CEE\)) introduced in [\textit{H. Çalışıcı} and \textit{E. Türkmen}, Georgian Math. J. 19, No. 2, 209--216 (2012; Zbl 1254.16002)] and so a generalization of \textit{H. Zöschinger}'s modules with the properties (E) and (EE) given in [Math. Scand. 35, 267--287 (1975; Zbl 0299.13006)]. We investigate various properties of these modules along with examples. In particular we prove these: (1) a module \(M\) has the property (\(\delta\)-\(CEE\)) if and only if every submodule of \(M\) has the property (\(\delta\)-\(CE\)); (2) direct summands of a module that has the property (\(\delta\)-\(CE\)) also have the property (\(\delta\)-\(CE\)); (3) each factor module of a module that has the property (\(\delta\)-\(CE\)) also has the property (\(\delta\)-\(CE\)) under a special condition; (4) every module with composition series has the property (\(\delta\)-\(CE\)); (5) over a \(\delta\)-\(V \)-ring a module \(M\) has the property (\(\delta\)-\(CE\)) if and only if \(M\) is cofinitely injective; (6) a ring \(R\) is \(\delta\)-semiperfect if and only if every left \(R\)-module has the property (\(\delta\)-\(CE\)).