A search for direct top squark pair production, in final states with one isolated electron or muon, multiple jets and missing transverse momentum, is presented. The analysis is performed using data from proton-proton collisions, which were collected in 2015 and 2016 at a center-of-mass energy of $\sqrt{s}=13~\rm{TeV}$ by the ATLAS collaboration, corresponding to an integrated luminosity of $36.1~\rm{fb^{−1}}$. Only pure bino LSP decay scenarios, where a $\tilde{t}_{1}$ particle decays into a top quark $t$ and a neutralino $(\tilde{\chi}^{0}_{1})$ are reported. Additional signatures are considered, including three-body and four-body decays where the top quarks or the $W$ bosons are produced off-shell, depending on the mass splitting between $\tilde{t}_{1}$ and $\tilde{\chi}^{0}_{1}$. No significant excesses in data compared to the expectations from Standard Model processes are observed, hence the results are interpreted as exclusion limits at 95 confidence level on the mass of the top squark and the neutralino.