A search for direct $\it{CP}$ violation in the Cabibbo-suppressed decay $D_{s}^{+}\rightarrow K^{-}K^{+}K^{+}$ and in the doubly Cabibbo-suppressed decay $D^{+}\rightarrow K^{-}K^{+}K^{+}$ is reported. The analysis is performed with data collected by the LHCb experiment in proton-proton collisions at a centre-of-mass energy of 13 TeV corresponding to an integrated luminosity of 5.6 $\textrm{fb}^{-1}$. The search is conducted by comparing the $D^+_{(s)}$ and $D^-_{(s)}$ Dalitz-plot distributions through a model-independent binned technique, based on fits to the $K^{-}K^{+}K^{+}$ invariant-mass distributions, with a total of 0.97 (1.27) million $D_{s}^{+}$ ($D^{+}$) signal candidates. The results are given as $p$-values for the hypothesis of $\it{CP}$ conservation and are found to be 13.3% for the $D_{s}^{+}\rightarrow K^{-}K^{+}K^{+}$ decay and 31.6% for the $D^{+}\rightarrow K^{-}K^{+}K^{+}$ decay. No evidence for $\it{CP}$ violation is observed in these decays.