Utilizing the pattern found on the Pb quantum islands formed above Si(111)–Pb surfaces, two-dimensional Ag nanoclusters can be successfully grown via the self-organized mechanism. Not only are Ag nanoclusters in the planar formed, but also naturally develop into distinct sizes and shapes. Those nanoclusters with enhanced stability are named magic Ag nanopucks. In the present work, scanning tunneling microscopy (STM) is employed to study the thermal stability strength of Ag nanopucks. By slowly raising the sample temperature, the physical stability is systematically demonstrated. Furthermore, the relative stability of magic Ag nanoclusters is also examined. In addition, the surrounding effect on the stability is also of our interest in the present work. For the purpose of realizing this effect, two different deposition coverage were chosen to deposit onto Pb quantum islands.