The physical and electronic properties of single-walled carbon nanotubes (SWCNTs) are determined by their chirality. The chirality selection mechanism in SWCNT growth is not fully understood. In this study, the interaction between near-armchair (n,5), where n = 6, 7, 8, and 9, zigzag (9,0), and armchair (5,5) nanotubes and a fully relaxed Ni-55 metal cluster during the early stage of growth is studied by density functional theory calculations. We found that kink sites at the end edge of (n,5) nanotubes are more reactive than other sites based on the charge transfer analysis at the Ni-C interface. The frontier orbitals of the (6,5) and (7,5) caps are localized on their kink-step sites, which stretch outward from the carbon cap surface, having typical 2p(z) orbital feature of carbon atom with high reactivity. Such favorable frontier orbital spatial orientation and location is ideal to incorporate more carbon species. These reactive sites may lead to the faster growth rate, resulting in the chirality selectivity toward the (6,5) and (7,5) nanotubes. In contrast, the frontier orbitals of (8,5) and (9,5) caps spread over the entire carbon cap surface. Adding carbon species at these sites may lead to the chirality change or formation of other carbon structures. Our results showed that the spatial distribution and orientation of frontier orbitals is useful in explaining the chiral selectivity. Engineering catalyst clusters to control these reactive sites has high potential to further improve chirality control in SWCNT synthesis.