  Water vapor (H$_{2}$O) is one of the brightest molecular emitters after
carbon monoxide (CO) in galaxies with high infrared (IR) luminosity, and allows
us to investigate the warm dense phase of the interstellar medium (ISM) where
star formation occurs. However, due to the complexity of its radiative
spectrum, H$_{2}$O is not frequently exploited as an ISM tracer in distant
galaxies. Therefore, H$_{2}$O studies of the warm and dense gas at high-$z$
remains largely unexplored. In this work we present observations conducted with
the Northern Extended Millimeter Array (NOEMA) toward three $z>6$ IR-bright
quasars J2310+1855, J1148+5251, and J0439+1634 targeted in their multiple
para-/ortho-H$_{2}$O transitions ($3_{12}-3_{03}$, $1_{11}-0_{00}$,
$2_{20}-2_{11}$, and $4_{22}-4_{13}$), as well as their far-IR (FIR) dust
continuum. By combining our data with previous measurements from the literature
we estimate dust masses and temperatures, continuum optical depths, IR
luminosities, and the star-formation rates from the FIR continuum. We model the
H$_{2}$O lines using the MOLPOP-CEP radiative transfer code and find that water
vapor lines in our quasar host galaxies are primarily excited in warm dense
(gas kinetic temperature and density of $T_{\rm kin} = 50\,{\rm K}$, $n_{\rm
H_{2}}\sim 10^{4.5}-10^{5}\,{\rm cm^{-3}}$) molecular medium with water vapor
column density of $N_{\rm H_{2}O}\sim 2\times10^{17}-3\times10^{18}\,{\rm
cm^{-3}}$. High-$J$ H$_{2}$O lines are mainly radiatively pumped by the intense
optically-thin far-IR radiation field associated with a warm dust component
with temperatures of $T_{\rm dust}\sim 80-190\,{\rm K}$ that account for
$<5-10\%$ of the total dust mass. Our results are in agreement with
expectations based on the H$_{2}$O spectral line energy distribution of local
and high-$z$ ultra-luminous IR galaxies and AGN. [abridged]
