A search for flavor-changing neutral-current couplings between a top quark, an up or charm quark, and a Formula Presented boson is presented, using proton-proton collision data at Formula Presented collected by the ATLAS detector at the Large Hadron Collider. The analyzed data set corresponds to an integrated luminosity of Formula Presented. The search targets both single-top-quark events produced as Formula Presented (with Formula Presented, Formula Presented) and top-quark-pair events, with one top quark decaying through the Formula Presented channel. The analysis considers events with three leptons (electrons or muons), a Formula Presented-tagged jet, possible additional jets, and missing transverse momentum. The data are found to be consistent with the background-only hypothesis and 95% confidence-level limits on the Formula Presented branching ratios, assuming only tensor operators of the Standard Model effective field theory framework contribute to the Formula Presented vertices. These are Formula Presented (Formula Presented) for Formula Presented (Formula Presented) for a left-handed Formula Presented coupling, and Formula Presented (Formula Presented) in the case of a right-handed coupling. These results are interpreted as 95% CL upper limits on the strength of the corresponding couplings, yielding limits for Formula Presented and Formula Presented (Formula Presented and Formula Presented) of 0.15 (0.16), and limits for Formula Presented and Formula Presented (Formula Presented and Formula Presented) of 0.22 (0.21), assuming a new-physics energy scale Formula Presented of 1 TeV.