************************************************************************* 62088 We propose ASCA observations of G349.7+0.2 and G344.7-0.1, two Galactic SNRs recently detected in an ASCA Galactic Plane survey project. The high absorption for each remnant indicates that the distances are large. The high energy response provided by ASCA is thus crucial in peering deep into the Galaxy to observe these objects. G349.7+0.2 has an extremely high radio surface brightness, probably indicating that it is a young ejecta-dominated or plerionic SNR. The ASCA survey data show clear emission lines and a hint of a hard spectral tail. G344.7-0.1 has a radio morphology suggestive of a composite remnant although the X-ray emission appears primarily thermal. We propose deeper observations of these SNRs in order to obtain sufficient counts to carry out detailed spectral investigations. *************************************************************************