  (abridged) The first unidentified very high energy gamma ray source (TeV
J2032+4130) in the Cygnus region has been the subject of intensive search for a
counterpart source at other wavelengths. A deep ($\approx 50$ ksec) exposure of
TeV J2032+4130 with \textit{XMM-Newton} has been obtained. The contribution of
point sources to the observed X-ray emission from TeV J2032+4130 is subtracted
from the data. The point-source subtracted X-ray data are analyzed using blank
sky exposures and regions adjacent to the position of TeV J2032+4130 in the
field of view covered by the XMM-Newton telescopes to search for diffuse X-ray
emission. An extended X-ray emission region with a full width half maximum
(FWHM) size of $\approx 12$ arc min is found. The centroid of the emission is
co-located with the position of TeV J2032+4130.The energy spectrum of the
emission coinciding with the position and extension of TeV J2032+4130 can be
modeled by a power-law model with a photon index
$\Gamma=1.5\pm0.2_\mathrm{stat}\pm0.3_\mathrm{sys}$ and an energy flux
integrated between 2 and 10 keV of $f_{2-10 \mathrm{keV}} \approx 7\cdot
10^{-13}$ ergs/(cm$^2$ s) which is lower than the very high energy gamma-ray
flux observed from TeV J2032+4130. We conclude that the faint extended X-ray
emission discovered in this observation is the X-ray counterpart of TeV
J2032+4130. Formally, it can not be excluded that the extended emission is due
to an unrelated population of faint, hot ($k_BT\approx 10$ keV) unresolved
point-sources which by chance coincides with the position and extension of TeV
J2032+4130. We discuss our findings in the frame of both hadronic and leptonic
gamma-ray production scenarios.
