  This article reports the results of X-ray studies of the extended TeV
$\gamma$-ray source VER J2019+368. Suzaku observations conducted to examine
properties of the X-ray pulsar wind nebula (PWN) around PSR J2021+3651 revealed
that the western region of the X-ray PWN has a source extent of $15' \times
10'$ with the major axis oriented to that of the TeV emission. The PWN-west
spectrum was closely fitted by a power-law for absorption at $N({\rm H}) =
(8.2^{+1.3}_{-1.1}) \times 10^{21}~{\rm cm^{-2}}$ and a photon index of $\Gamma
= 2.05\pm0.12$, with no obvious change in the index within the X-ray PWN. The
measured X-ray absorption indicates that the distance to the source is much
less than $10~{\rm kpc}$ inferred by radio data. Aside from the PWN, no
extended emission was observed around PSR J2021+3651 even by Suzaku. Archival
data from the XMM-Newton were also analyzed to complement the Suzaku
observations, indicating that the eastern region of the X-ray PWN has a similar
spectrum ($N(\rm H)=(7.5 \pm 0.9) \times 10^{21}~{\rm cm^{-2}}$ and
$\Gamma=2.03 \pm 0.10$) and source extent up to at least $12'$ along the major
axis. The lack of significant change in the photon index and the source extent
in X-ray are used to constrain the advection velocity or the diffusion
coefficient for accelerated X-ray-producing electrons. A mean magnetic field of
${\sim}3~\mu{\rm G}$ is required to account for the measured X-ray spectrum and
reported TeV $\gamma$-ray spectrum. A model calculation of synchrotron
radiation and inverse Compton scattering was able to explain ${\sim}80\%$ of
the reported TeV flux, indicating that the X-ray PWN is a major contributor of
VER J2019+368.
