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 $Gamma = 2.05pm0.12$, 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.
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