We present results obtained from near-infrared JHK spectroscopic observations of novae V2491 Cygni and V597 Puppis in the early declining phases of their 2007 and 2008 outbursts, respectively. In both objects, the spectra displayed emission lines of Hi, Oi, He i and Ni. In V597 Pup, the Hei lines were found to strengthen rapidly with time. Based on the observed spectral characteristics, both objects are classified as He/N novae. We have investigated the possibility of V2491 Cyg being a recurrent nova as has been suggested. By studying the temporal evolution of the linewidths in V2491 Cyg, it appears unlikely that the binary companion is a giant star with heavy wind as in recurrent novae of the RS Oph type. Significant deviations from that of recombination case B conditions are observed in the strengths of the Hi lines. This indicates that the Hi lines, in both novae, are optically thick during the span of our observations. The slope of the continuum spectra in both cases was found to have a λ–(3–3.5) dependence which deviates from a Rayleigh-Jeans spectral distribution. Both novae were detected in the post-outburst supersoft X-ray phase; V2491 Cyg being very bright in X-rays has been the target of several observations. We discuss and correlate our infrared observations with the observed X-ray properties of these novae.