We report results of an X-ray study of the supernova remnant (SNR) G344.7-0.1 and the point-like X-ray source located at the geometrical center of the SNR radio structure. The morphology and spectral properties of the remnant and the central X-ray point-like source were studied using data from the XMM-Newton and Chandra satellites. Archival radio data and infrared Spitzer observations at 8 and 24µm were used to compare and study its multi-band properties at different wavelengths. The XMM-Newton and Chandra observations reveal that the overall X-ray emission of G344.7-0.1 is extended and correlates very well with regions of bright radio and infrared emission. The X-ray spectrum is dominated by prominent atomic emission lines. These characteristics suggest that the X-ray emission originated in a thin thermal plasma, whose radiation is represented well by a plane-parallel shock plasma model (PSHOCK). Our study favors the scenario in which G344.7-0.1 is a 6x103 year old SNR expanding in a medium with a high density gradient and is most likely encountering a molecular cloud on the western side. In addition, we report the discovery of a soft point-like X-ray source located at the geometrical center of the radio SNR structure. The object presents some characteristics of the so-called compact central objects (CCO). However, its neutral hydrogen absorption column (NH) is inconsistent with that of the SNR. Coincident with the position of the source, we found infrared and optical objects with typical early-K star characteristics. The X-ray source may be a foreground star or the CCO associated with the SNR. If this latter possibility were confirmed, the point-like source would be the farthest CCO detected so far and the eighth member of the new population of isolated and weakly magnetized neutron stars.