In massively multiplexed spectroscopic surveys like the Dark Energy Spectroscopic Instrument (DESI) survey, a serendipitous fraction of integrated spectra will include the light of both the targeted foreground galaxy and a gravitationally lensed background galaxy. These so-called spectroscopic lenses are important because they can help place constraints on the dark matter properties of the foreground galaxy, elucidate the physical properties of the distant background galaxy, and, through a statistical analysis, constrain cosmological parameters. In 2021-2026, DESI will obtain spectra for more than 30 million galaxies at z < 1.5, including a flux-limited sample (r < 20) of bright galaxies at < z > =0.3. We use simulated spectra to constrain our ability to identify spectroscopic lenses in DESI survey data and to measure the redshift of the background galaxy as a function of observing conditions and the relative brightness of the background galaxy. Our effort lays the foundation for running a dedicated search for spectroscopic lenses in DESI.