An element $a$ in a ring R is called left morphic if $$R/Ra{simeq_-}1(a)$$. A ring is called left morphic if every element is left morphic. In this paper, an element $a$ in a ring R is called left ${pi}$-morphic (resp. left G-morphic) if there exists a positive number $n$ such that $a^n$ (resp. $a^n{ eq}0$) is left morphic. A ring R is called left ${pi}$-morphic (resp. left G-morphic) if every element is left ${pi}$-morphic (resp. left G-morphic). The Morita invariance of left ${pi}$-morphic (resp. left G-morphic) rings is discussed. Several relevant properties are proved. In particular, it is shown that a left Noetherian ring R with $M_4(R)$ left G-morphic or $M_2(R)$ left morphic is QF. Some known results of left morphic rings are extended to left G-morphic rings and left ${pi}$-morphic rings.