Abstract We present results from our X-ray analysis of a systematic search for triple active galactic nuclei (AGNs) in nearby (z < 0.077) triple galaxy mergers. We analyze archival Chandra observations of seven triple galaxy mergers with baymax (Bayesian Analysis of Multiple AGNs in X-rays), fitting each observation with single, dual, and triple X-ray point-source models. In doing so, we conclude that one triple merger has one X-ray point source (SDSS J0858+1822, although it is unlikely to be an AGN), five triple mergers are likely composed of two X-ray point sources (NGC 3341, SDSS J1027+1749, SDSS J1631+2352, SDSS J1708+2153, and SDSS J2356−1016), and one system is composed of three X-ray point sources (SDSS J0849+1114). By fitting the individual X-ray spectra of each point source, we analyze the 2−7 keV luminosities, as well as the levels of obscuration associated with each potential AGN. We find that 4/5 dual X-ray point-source systems have primary and secondary point sources with bright X-ray luminosities (L 2−7kev > 1040 erg s−1), possibly associated with four new undetected dual AGNs. The dual and triple-point-source systems are found to have physical separations between 3 and 9 kpc and flux ratios between 2 × 10−3 and 0.84. A multiwavelength analysis to determine the origin of the X-ray point sources discovered in this work is presented in our companion paper (Foord et al. 2020c).