Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module. In this paper, we introduce a class of modules which is analogous to that of Goldie$^*$-lifting and principally Goldie$^*$-lifting modules. The module~$M$ is called \textit {principally} $\mathcal {G}^*$-\nobreak $\delta $-\textit {lifting} if, for any $m\in M$, there exists a direct summand $N$ of $M$ such that $mR$ is $\beta ^*_{\delta }$-equivalent to $N$. We also introduce a generalization of Goldie$^*$-supplemented modules, namely, a module $M$ is said to be \textit {principally} $\mathcal {G}^*$-$\delta $-\textit {supplemented} if, for any $m\in M$, there exists a $\delta $-supplement~$N$ in~$M$ such that $mR$ is $\beta ^*_{\delta }$-equivalent to~$N$. We prove that some results of principally $\mathcal {G}^*$-lifting modules and Goldie$^*$-lifting modules can be extended to principally $\mathcal {G}^*$-\nobreak $\delta $-lifting modules for this general setting. Several properties of these modules are given, and it is shown that the class of principally $\mathcal {G}^*$-\nobreak $\delta $-lifting modules lies between the classes of principally $\delta $-lifting modules and principally $\mathcal {G}^*$-$\delta $-supplemented modules.