A search for leptoquarks produced singly and in pairs in proton-proton collisions is presented. We consider the leptoquark (LQ) to be a scalar particle of charge −1/3e coupling to a top quark plus a tau lepton (tτ) or a bottom quark plus a neutrino (bν), or a vector particle of charge +2/3e, coupling to tν or bτ. These choices are motivated by models that can explain a series of anomalies observed in the measurement of B meson decays. In this analysis the signatures tτνb and tτν are probed, using data recorded by the CMS experiment at the CERN LHC at s=13 TeV and that correspond to an integrated luminosity of 137 fb−1. These signatures have not been previously explored in a dedicated search. The data are found to be in agreement with the standard model prediction. Lower limits at 95% confidence level are set on the LQ mass in the range 0.98–1.73 TeV, depending on the LQ spin and its coupling λ to a lepton and a quark, and assuming equal couplings for the two LQ decay modes considered. These are the most stringent constraints to date on the existence of leptoquarks in this scenario.