Recently, deviations in flavor observables of B→D(∗)τν have been shown between the predictions in the Standard Model and the experimental results reported by BABAR, Belle, and LHCb collaborations. One of the solutions to this anomaly is obtained in a class of leptoquark model with a scalar leptoquark boson S1, which is a SU(3)c triplet and SU(2)L singlet particle with -1/3 hypercharge interacting with a quark-lepton pair. With well-adjusted couplings, this model can explain the anomaly and be compatible with all flavor constraints. In such a case, the S1 boson can be pair-produced at CERN's Large Hadron Collider (LHC) and subsequently decay as S1∗→tτ, bντ, and cτ. This paper explores the current 8 and 13 TeV constraints, as well as the detailed prospects at 14 TeV, of this flavor-motivated S1 model. From the current available 8 and 13 TeV LHC searches, we obtain constraints on the S1 boson mass for MS1<400-640 GeV depending on values of the leptoquark couplings to fermions. Then we study future prospects for this scenario at the 14 TeV LHC using detailed cut analyses and evaluate exclusion and discovery potentials for the flavor-motivated S1 leptoquark model from searches for the (bν)(bν) and (cτ)(cτ) final states. In the latter case, we consider several scenarios for the identification of charm jets. As a result, we find that the S1 leptoquark origin of the B→D(∗)τν anomaly can be probed with MS1600/800 GeV at the 14 TeV LHC with L=300/3000 fb-1 of accumulated data. One can also see that the 14 TeV LHC run II with L=300 fb-1 can exclude the S1 leptoquark boson up to MS1∼0.8 TeV at 95% confidence level, whereas a future 14 TeV LHC with L=3000 fb-1 data has a potential to discover the S1 leptoquark boson with its mass up to MS1∼1.1 TeV with over 5σ significance, from the (bν)(bν) and/or (cτ)(cτ) searches. © 2016 American Physical Society.