Aims. 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. Methods. 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 24 $mu$m were used to compare and study its multi-band properties at different wavelengths. Results. 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 6 x 10^3 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 (N_{H}) 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.