Summary: A module \(M\) is called \textit{FI-extending} if every fully invariant submodule of \(M\) is essential in a direct summand of \(M\). In this work, we define a module \(M\) to be \textit{generalized FI-extending (GFI-extending)} if for any fully invariant submodule \(N\) of \(M\), there exists a direct summand \(D\) of \(M\) such that \(N \leq D\) and that \(D/N\) is singular. The classes of FI-extending modules and singular modules are properly contained in the class of GFI-extending modules. We first develop basic properties of this newly defined class of modules in the general module setting. Then, the GFI-extending property is shown to carry over to matrix rings. Finally, we show that the class of GFI-extending modules is closed under direct sums but not under direct summands. However, it is proved that direct summands are GFI-extending under certain restrictions.