The nature of the metallicity gradient inside the solar circle (R_GC < 8 kpc) is poorly understood, but studies of Cepheids and a small sample of open clusters suggest that it steepens in the inner disk. We investigate the metallicity gradient of the inner disk using a sample of inner disk open clusters that is three times larger than has previously been studied in the literature to better characterize the gradient in this part of the disk. We used the Gaia-ESO Survey (GES) [Fe/H] values and stellar parameters for stars in 12 open clusters in the inner disk from GES-UVES data. Cluster mean [Fe/H] values were determined based on a membership analysis for each cluster. Where necessary, distances and ages to clusters were determined via comparison to theoretical isochrones. The GES open clusters exhibit a radial metallicity gradient of -0.10+-0.02 dex/kpc, consistent with the gradient measured by other literature studies of field red giant stars and open clusters in the range R_GC ~ 6-12 kpc. We also measure a trend of increasing [Fe/H] with increasing cluster age, as has also been found in the literature. We find no evidence for a steepening of the inner disk metallicity gradient inside the solar circle as earlier studies indicated. The age-metallicity relation shown by the clusters is consistent with that predicted by chemical evolution models that include the effects of radial migration, but a more detailed comparison between cluster observations and models would be premature.