This article reports the results of X-ray studies of the extended TeV γ-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 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 and a photon index of , 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 the 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 ( and ) and source extent up to at least 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 is required to account for the measured X-ray spectrum and reported TeV γ-ray spectrum. A model calculation of synchrotron radiation and inverse Compton scattering was able to explain of the reported TeV flux, indicating that the X-ray PWN is a major contributor of VER J2019+368.