We present the results of deep ASCA observations of two shell-like radio supernova remnants (SNRs) located in the direction to the Galactic center (GC) region. Unlike the radio morphology, G 359.1-0.5 shows center-filled X-rays with prominent K alpha lines of He-like silicon and H-like sulfur. The plasma requires at least two temperature components: a silicon-dominated cool plasma of 0.6 keV temperature and a sulfur-dominated higher temperature plasma of 4.4 keV. Because the absorption column is 6* 1022 H cm-2, this SNR would be near to the GC. The spherical plasma is attributable to supernova ejecta with the total mass of Si and S being about 0.1 Mo and 0.3 Mo, respectively. X-rays from G 359.0-0.9 trace the partial shell structure of the radio emission. The spectrum is well fitted to a single-temperature plasma of 0.4 keV with a non-solar abundance of magnesium or iron. Because the absorption column is not very large, 1.8* 1022 H cm-2, G 359.0-0.9 would be in front of the GC region. The total supernova energy, interstellar density near to the X-ray emitting shell and age of the SNR are estimated to be 1.2*1051 erg, 0.5 cm-3, and 1.8*104 yr, respectively. We also discuss possible implications on the origin of the large-scale hot plasma surrounding the GC.