We report on Suzaku observations of large-scale X-ray structures possibly related with the Fermi Bubbles obtained in 2013 with a total duration of ~ 80 ks. The observed regions were the: (i) northern cap (N-cap; l ~ 0 deg, 45 deg < b < 55 deg) seen in the Mid-band (1.7-4.0 keV) map recently provided by MAXI-SSC and (ii) southeast claw (SE-claw; l ~ 10 deg, -20 deg < b < -10 deg) seen in the ROSAT all-sky map and MAXI-SSC Low-band (0.7-1.7 keV) map. In each region, we detected diffuse X-ray emissions which are represented by a three component plasma model consisting of an unabsorbed thermal component (kT ~ 0.1 keV) from the Local Bubble, absorbed kT = 0.30+/-0.05 keV emission representing the Galactic Halo, and a power-law component due to the isotropic cosmic X-ray background radiation. The emission measure of the GH component in the SE-claw shows an excess by a factor of ~ 2.5 over the surrounding emission at 2 deg away. We also found a broad excess in the 1.7-4.0 keV count rates across the N-cap after compiling other archival data from Suzaku and Swift. The spectral stacking analysis of the N-cap data indicates the presence of another thermal component with kT = 0.70 (+0.22,-0.11) keV. The temperature of kT ~ 0.3 keV of the Galactic Halo is higher than the ubiquitous value of kT ~ 0.2 keV near the Fermi Bubbles, and can be even higher (~ 0.7 keV). We discuss our findings in the context of bubble-halo interaction.