We present the results of ROSAT PSPC X-ray observations of NGC 4125 and NGC 3610. These two galaxies are among the best representatives of a newly recognized class of elliptical galaxies that may be the product of recent mergers, NGC 4125 shows two plumes or disks crossing at right angles; NGC 3610 has the richest fine structure of all elliptical galaxies cataloged by Schweizer and Seltzer. Our X-ray observations show that these galaxies do not retain large gaseous halos, contrary to the suggested link between merging and X-ray brightness. Their X-ray luminosities (0.1-2 keV) are in the range 10(40-41) ergs s(-1), low compared with those of hot gas-rich elliptical galaxies. The X-ray to optical ratio of NGC 3610 is in the range of those of X-ray faint elliptical galaxies, consistent with the value found for the bulge of M31, where the X-ray emission is dominated by low-mass binaries; its X-ray spectrum is also comparable with those of X-ray faint elliptical galaxies, presenting significant very soft excess emission. The X-ray to optical ratio of NGC 4125 is larger and might suggest the presence of some hot interstellar medium. This suggestion is supported by the spectrum of NGC 4125, which shows evidence of Fe L emission. This spectrum can be fitted with either a low-abundance single-temperature Raymond model or with a two-temperature model with solar abundances. Further work is needed to firmly resolve the question of gaseous versus stellar X-ray emission in NGC 4125.