Somatostatin analogues are useful pharmaceuticals in peptide receptor radionuclide therapy. In previous studies, we analyzed a new bicyclic somatostatin analogue (BCS) in connection with Cu(II) ions. Two characteristic sites were present in the peptide chain: the receptor- and the metal-binding site. We have already shown that this ligand can form very stable imidazole complexes with the metal ion. In this work, our aim was to characterize the intramolecular interaction that occurs in the peptide molecule. Therefore, we analyzed the coordination abilities of two cyclic ligands, i.e., P1 only with the metal binding site and P2 with both sites, but without the disulfide bond. Furthermore, we used magnetic circular dichroism (MCD) spectroscopy to better understand the coordination process. We applied this method to analyze spectra of P1, P2, and BCS, which we have described previously. Additionally, we analyzed the MCD spectra of P3 ligand, which has only the receptor binding site in its structure. We have unequivocally shown that the presence of the Phe-Trp-Lys-Thr motif and the disulfide bond significantly increases the metal binding efficiency.