Abstract: A search is presented for the pair production of heavy vector-like T quarks, primarily targeting the T quark decays to a W boson and a b-quark. The search is based on 36: 1 fb(-1) of pp collisions at root s = 13TeV recorded in 2015 and 2016 with the ATLAS detector at the CERN Large Hadron Collider. Data are analysed in the lepton-plus-jets final state, including at least one b-tagged jet and a large-radius jet identified as originating from the hadronic decay of a high-momentum W boson. No significant deviation from the Standard Model expectation is observed in the reconstructed T mass distribution. The observed 95% confidence level lower limit on the T mass are 1350 GeV assuming 100% branching ratio to Wb. In the SU(2) singlet scenario, the lower mass limit is 1170 GeV. This search is also sensitive to a heavy vector-like B quark decaying to Wt and other final states. The results are thus reinterpreted to provide a 95% con fidence level lower limit on the B quark mass at 1250 GeV assuming 100% branching ratio to Wt; in the SU(2) singlet scenario, the limit is 1080 GeV. Mass limits on both T and B production are also set as a function of the decay branching ratios. The 100% branching ratio limits are found to be applicable to heavy vector-like Y and X production that decay to Wb and Wt, respectively.