We study the prospects of measuring the CP property of the Higgs ($h$) coupling to tau leptons using the vector boson fusion (VBF) production mode at the high-luminosity LHC. Utilizing the previously proposed angle between the planes spanned by the momentum vectors of the $(pi^+pi^0)$ and $(pi^- pi^0)$ pairs originating in $tau^pm$ decays as the CP-odd observable, we perform a detailed Monte Carlo analysis, taking into account the relevant standard model backgrounds, as well as detector resolution effects. We find that excluding a pure CP-odd coupling hypothesis requires $mathcal{O}(400 {~rm fb}^{-1})$ luminosity at the 14 TeV LHC, and values of the CP-mixing angle larger than about $25^circ$ can be excluded at $95%$ confidence level using $3 {~rm ab}^{-1}$ data. It is observed that the uncertainty in the angular resolution of the neutral pion momenta does not constitute a significant hurdle. Achieving a signal to background ratio ($S/B$) close to one, while keeping a high enough signal yield required to study the angular distributions selects out VBF as a promising mode to probe the CP nature of the $htautau$ coupling, with gluon fusion suffering from a low $S/B$, and the $W^pm h/Zh$ mode (with leptonically decaying $W^pm /Z$) having a much smaller signal rate.
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