Context : 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 .
In particular , observations in radio and X-rays are important to trace a population of non-thermal electrons .

Aims : A deep ( \approx 50 ksec ) exposure of TeV J2032+4130 with XMM-Newton has been obtained .
The large collection area and the field of view of the X-ray telescopes on-board of XMM-Newton allow to search for faint extended X-ray emission possibly linked to TeV J2032+4130 .

Methods : 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 .

Results : 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 angular extension of the X-ray emission region is slightly smaller than the angular size of TeV J2032+4130 ( FWHM= 14 \pm 3 arc min ) .
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 \pm 0.2 _ { \mathrm { stat } } \pm 0.3 _ { \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 .
The energy flux detected from the extended emission region is about a factor of two smaller than the summed contribution of the point sources present .
The energy spectrum can also be fit with a thermal emission model from an ionized plasma with a temperature k _ { B } T \approx 10 keV .

Conclusions : 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 _ { B } T \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 .