The kinetic properties of macrocyclic ligand 2-[1,4,7,10–tetraazacyclododecane-1-yl]-ethanethiol (L1, DOSH) were studied with Cu(II) and Zn(II) metal ions, using conventional molecular absorption spectroscopy in pH regions 2.5–4.7 and 4.3–6.0 for Cu(II) and Zn(II), respectively. The obtained partial rate constants, measured at 25 °C and ionic strength I = 0.1 M (KCl), are slightly lower than for parent cyclen (1,4,7,10-tetraazacyclododecane) macrocyclic ligand supporting the hypothesis that the thiol pendant arm is not directly involved in the formation of these metal complexes. The study of the dissociation of the copper(II)-L1 complex in the presence of perchloric acid and at higher temperatures (75–95 °C) showed that this complex is more kinetically inert than the copper(II) complex with cyclen, indicating that the presence of the thiol pendant arm plays a key role as protecting group of copper(II) ion. On contrary, the dissociation of zinc(II) complex in presence of copper(II) ion measured at laboratory temperature (25 °C) and in 0.1 M-solution of hydrochloric acid is going via the metal-exchange reaction mechanism. Altogether, these findings show that L1 ligand presents favorable kinetic properties for further conjugation with biomolecules, aiming at their use in medicinal chemistry.