Abstract: As a top-factory, the LHC is performing a direct study of top-quark anomalous FCNC couplings, which are, however, correlated closely with the rare B- and K-meson decays. In this paper, we study the effects of anomalous tqZ (with q = u, c) couplings in the rare decays B-s,B-d -> mu(+)mu(-), B -> X-s nu(nu) over bar, B -> K(*)nu(nu) over bar, K+ -> pi(+)nu(nu) over bar, and K-L -> pi(0)nu(nu) over bar. With the up-to-date experimental bounds on the branching ratios of these channels, constraints on the left-handed anomalous couplings X-ct(L), and X-ut(L) are derived, respectively. With these low-energy constraints taken into account, we find that, for real couplings X-ct(L) and X-ut(L), the indirect upper bounds on B(t -> qZ) are much lower than that from the D0 collaboration, but are still compatible with the 5 sigma discovery potential of ATLAS with an integrated luminosity of 10 fb(-1). With refined measurements to be available at the LHCb, the future super-B factories, the NA62 at CERN, and the KOTO at J-PARC, closer correlations between the t -> qZ and the rare B- and K-meson decays are expected in the near future, which will be helpful for the searches of thu e top-quark FCNC decays at the LHC.