We present Suzaku X-ray observations along two edge regions of the Fermi Bubbles , with eight \simeq 20 ksec pointings across the northern part of the North Polar Spur ( NPS ) surrounding the north bubble and six across the southernmost edge of the south bubble . After removing compact X-ray features , diffuse X-ray emission is clearly detected and is well reproduced by a three-component spectral model consisting of unabsorbed thermal emission ( temperature kT \simeq 0.1 keV ) from the Local Bubble ( LB ) , absorbed kT \simeq 0.3 keV thermal emission related to the NPS and/or Galactic Halo ( GH ) , and a power-law component at a level consistent with the cosmic X-ray background . The emission measure ( EM ) of the 0.3 keV plasma decreases by \simeq 50 \% toward the inner regions of the north-east bubble , with no accompanying temperature change . However , such a jump in the EM is not clearly seen in the south bubble data . While it is unclear if the NPS originates from a nearby supernova remnant or is related to previous activity within/around the Galactic Center , our Suzaku observations provide evidence suggestive of the latter scenario . In the latter framework , the presence of a large amount of neutral matter absorbing the X-ray emission as well as the existence of the kT \simeq 0.3 keV gas can be naturally interpreted as a weak shock driven by the bubbles ’ expansion in the surrounding medium , with velocity v _ { exp } \sim 300 km s ^ { -1 } ( corresponding to shock Mach number \mathcal { M } \simeq 1.5 ) , compressing the GH gas to form the NPS feature . We also derived an upper limit for any non-thermal X-ray emission component associated with the bubbles and demonstrate , that in agreement with the findings above , the non-thermal pressure and energy estimated from a one-zone leptonic model of its broad-band spectrum , are in rough equilibrium with that of the surrounding thermal plasma .