A search is made for the highly-suppressed B meson decays $B^{+}\rightarrow K^{+}K^{+}\pi^{-}$ and $B^{+}\rightarrow \pi^{+}\pi^{+}K^{-}$ using a data sample corresponding to an integrated luminosity of 3.0 $fb^{-1}$ collected by the LHCb experiment in proton-proton collisions at centre-of-mass energies of 7 and 8 TeV. No evidence is found for the decays, and upper limits at 90\% confidence level are determined to be $\mathcal{B}(B^{+}\rightarrow K^{+}K^{+}\pi^{-}) < 1.1\times 10^{-8}$ and $\mathcal{B}(B^{+}\rightarrow \pi^{+}\pi^{+}K^{-}) < 4.6\times 10^{-8}$.