Following the identification of the optical afterglow (Quirola-Vasquez et al., GCN 37319) of GRB 240821A (Cangemi et al., GCN 37220; He et al., GCN 37226; Murphy et al., GCN 37232; Dalessi & Meegan, GCN 37239), we obtained 4x1200 s of observations with the X-shooter instrument at the ESO VLT UT3 (mid time 2024 Aug 30.3578 UT, that is, 8.8 days after the trigger). These observations reveal a continuum across the spectral range 4000-20000 AA, and a single clear emission line at 8120 AA. While the presence of a substantial continuum blueward of 8120 AA rules out that this line is Ly-alpha, it is difficult to conclusively determine a redshift value based on only a single line. We consider three possibilities below. A) The emission line could arise from the [O II] doublet (3726, 3729) at z = 1.18. In this scenario the other strong emission lines lie in the IR regime, where sensitivity is poorer and sky background higher. The observed spectral energy distribution of the host galaxy from the Legacy and VISTA surveys would also provide the best match at this redshift. However, the resolution of X-shooter should clearly resolve the two components of the doublet, and, while a very weak excess may be present at 8215 AA (the expected location of the bluer component of the doublet), the observed intensity ratio would be inconsistent with the expectations for either high or low density media. B) The line could be from [O III] (5007 AA) at z = 0.62. In this scenario, both [O III] 4959 and H-beta would lie in close proximity in the spectrum, although are expected to have lower flux, and hence poorer signal to noise than the observed line. There is no detection of any emission at the expected location of either. C) The line may come from H-alpha at z = 0.24. Since H-alpha may well be the brightest line within the host galaxy (especially for a short-GRB host), the non-detection of other emission features at the same redshift, but at bluer wavelengths is less problematic because they may be fainter and/or extinguished. This may also be supported by an apparent decrease in the galaxy continuum at ~4500 AA, which could be due to the Balmer break, although the signal to noise is poor in this region. The detection of a single line does not enable at the current time an unambiguous redshift determination, but does suggest it must be one of the above possibilities. We note that all of these lie outside the formal 1-sigma range of the photometric redshift determination from the Legacy survey (z = 0.488 +/- 0.074; Zhou et al. 2021, MNRAS, 501, 3309). Further analysis is ongoing, and we will update the community should this reveal a stronger conclusion regarding the burst redshift. We acknowledge expert help of the ESO observing staff at Paranal, in particular Thomas Rivinius and Matias Jones.