Taking into account noise from intrinsic ellipticities of source galaxies, in this paper, we study the peak statistics in weak-lensing convergence maps around clusters of galaxies and beyond. We emphasize how the noise peak statistics is affected by the density distribution of nearby clusters, and also how cluster-peak signals are changed by the existence of noise. These are the important aspects to be understood thoroughly in weak-lensing analyses for individual clusters as well as in cosmological applications of weak-lensing cluster statistics. We adopt Gaussian smoothing with the smoothing scale $\theta_G=0.5\hbox{ arcmin}$ in our analyses. It is found that the noise peak distribution near a cluster of galaxies depends sensitively on the density profile of the cluster. For a cored isothermal cluster with the core radius $R_c$, the inner region with $R\le R_c$ appears noisy containing on average $∼ 2.4$ peaks with $ν\ge 5$ for $R_c= 1.7\hbox{ arcmin}$ and the true peak height of the cluster $ν=5.6$, where $ν$ denotes the convergence signal to noise ratio. For a NFW cluster of the same mass and the same central $ν$, the average number of peaks with $ν\ge 5$ within $R\le R_c$ is $∼ 1.6$. Thus a high peak corresponding to the main cluster can be identified more cleanly in the NFW case. In the outer region with $R_c<R\le 5R_c$, the number of high noise peaks is considerably enhanced in comparison with that of the pure noise case without the nearby cluster. (abridged) Comment: 10 figures, ApJ in press