The stability of structure, spin, orbital magnetic moment and magnetic anisotropy energy of TM@Cu₁₂N₁₂ (TM = Mn, Fe, Co, Ni, Cu, Ru, Rh, Pd, Ir, Pt) are systematically investigated within the framework of the generalized gradient approximation with on-site coulomb repulsion density-functional theory (DFT-GGA+U). In the orbital moment and magnetic anisotropy energy (MAE) computation procedure, the spin-orbit coupling is considered and implemented. In this article, we mainly focus on the structure stability and tunable magnetism of the TM atom (Mn, Fe, Co, Ni, Cu, Ru, Rh, Pd, Ir, Pt) substituting the centre atom of icosahedron (ICO) Cu₁₃N₁₂ cluster, finally disclose the physics origin of the structure stability, change magnetism and larger MAE. The results show that the different TM atom doping makes the ICO structure of Cu₁₃N₁₂ cluster appears a tiny deformation. The stabilities of the clusters are evidently enhanced due to the formation of Cu—N and Cu—TM bond. In addition, the N-capped clusters more prefer to present a larger magnetic moment than the pure Cu₁₃ one. The magnetic environment of clusters is improved to varying degrees by doping different TM (TM = Mn, Fe, Co, Ni, Cu, Ru, Rh, Pd, Ir, Pt) atoms, which endows TM@Cu₁₂N₁₂ clusters with various magnetic properties. For instance, the doping of 3d atoms further enhances the spin magnetic moment of the clusters, the Mn, Fe and Co atoms replacing the centre atom of the ICO Cu₁₃N₁₂ generate 35, 32 and 33 giant moments, respectively. In light of the doping of 4d, 5d transition metal atoms, the orbital moments of the TM@Cu₁₂N₁₂ clusters do not increase evidently, but the MAE remarkably strengthens for the doping of Rh and Pt atoms, the MAE values reach to 15.34 meV/atom and 6.76 meV/atom for Rh@Cu₁₂N₁₂ and Pt@Cu₁₂N₁₂, respectively. The tunable magnetism of TM@Cu₁₂N₁₂ cluster provides promising applications in spintronics.