ABSTRACT In our previous works, we found absorbed thermal X-ray plasma with kT 0.3 keV observed ubiquitously near the edges of the Fermi bubbles and interpreted this emission as weakly shock-heated Galactic halo 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 < 20° and latitude 5° < 60°, covering the whole extent of the Fermi bubbles. We show that the plasma temperature is constant at kT 0.30 0.07 keV, while the emission measure (EM) varies by an order of magnitude, increasing toward the Galactic center (i.e., low ) with enhancements at the North Polar Spur (NPS), SE-claw, and NW-clump features. Moreover, the EM distribution of kT 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°) favors (ii), whereas that of the south (b < 0°) is rather close to (i), but a weak excess signature is clearly detected also in the south like NPS (South Polar Spur). Such an asymmetry, if due to the bubbles, cannot be fully understood only by the inclination of bubbles' axis against the Galactic disk normal, thus suggesting asymmetric outflow due to different environmental/initial conditions.