After the discovery of helicity-independent all-optical switching (HI-AOS)in amorphous GdFeCo using femtosecond laser pulse, significant efforts are made to understand the underlying mechanism in HI-AOS and also to utilize this phenomenon for ultra-high density data storage applications. HI-AOS has predominantly been observed in Gd-based ferrimagnetic alloys such as GdFeCo, GdCo, GdFe, etc. Efforts are required to extend the range of material which shows HI-AOS. With the help of atomistic spin dynamics calculations combined with two temperature model, we demonstrate the single-shot helicity-independent all-optical switching in amorphous TbxCo100−x ferrimagnet using femtosecond laser pulses. The value of x is varied from x = 18% to x = 36%, and a deterministic HI-AOS is observed for the composition range x = 20% to x = 32%. The threshold input laser energy is found to decrease with an increase in Tb concentration. Understanding the role of the damping parameter is crucial to understand the different coupling mechanism that governs HI-AOS. Further, an element specific damping is applied to both sublattices, and the damping constant of the Tb sublattice is kept at 0.05 while that of the Co sublattice is varied from 0.005 to 0.05. The results show the damping constant strongly influences the HI-AOS, and the higher damping constant is crucial for the HI-AOS. © 2023 Elsevier B.V.