In our previous works ( Kataoka et al . 2013 , Tahara et al . 2015 ) , we found absorbed thermal X-ray plasma with kT \simeq 0.3 keV observed ubiquitously near the edges of the Fermi bubbles and interpreted this emission as weakly shock-heated Galactic halo ( GH ) gas . Here we present a systematic and uniform analysis of archival Suzaku ( 29 pointings ; 6 newly presented ) and Swift ( 68 pointings ; 49 newly presented ) data within Galactic longitudes | l | < 20 ^ { \circ } and latitude 5 ^ { \circ } \lesssim | b | < 60 ^ { \circ } , covering the whole extent of the Fermi bubbles . We show that the plasma temperature is constant at kT \simeq 0.30 \pm 0.07 keV , while the emission measure ( EM ) varies by an order of magnitude , increasing toward the Galactic center ( i.e. , low |b| ) with enhancements at the north polar spur ( NPS ) , SE-claw and NW-clump features . Moreover , the EM distribution of kT \simeq 0.30 keV plasma is highly asymmetric in the northern and southern bubbles . Although the association of the X-ray emission with the bubbles is not conclusive , we compare the observed EM properties with simple models assuming ( i ) a filled halo without bubbles , whose gas density follows a hydrostatic isothermal model ( King profile ) and ( ii ) a bubble-in-halo in which two identical bubbles expand into the halo forming thick shells of swept halo gas . We argue that the EM profile in the north ( b > 0 ^ { \circ } ) favors ( ii ) , whereas that of the south ( b < 0 ^ { \circ } ) is rather close to ( i ) , but weak excess signature is clearly detected also in the south like NPS ( South Polar Spur ; SPS ) . Such an asymmetry , if due to the bubbles , can not be fully understood only by the inclination of bubbles ’ axis against the Galactic disk normal , thus suggesting asymmetric outflow due to different environmental/initial condition .