We present mid-infrared (MIR) 8-13 μm spectroscopy of the nuclear regions of the interacting galaxy Arp 299 (IC 694+NGC 3690) obtained with CanariCam (CC) on the 10.4 m Gran Telescopio Canarias (GTC). The high angular resolution (~0.''3-0.''6) of the data allows us to probe nuclear physical scales between 60 and 120 pc, which is a factor of 10 improvement over previous MIR spectroscopic observations of this system. The GTC/CC spectroscopy displays evidence of deeply embedded active galactic nucleus (AGN) activity in both nuclei. The GTC/CC nuclear spectrum of NGC 3690/Arp 299-B1 can be explained as emission from AGN-heated dust in a clumpy torus with both a high covering factor and high extinction along the line of sight. The estimated bolometric luminosity of the AGN in NGC 3690 is 3.2 ± 0.6 × 1044 erg s–1. The nuclear GTC/CC spectrum of IC 694/Arp 299-A shows 11.3 μm polycyclic aromatic hydrocarbon emission stemming from a deeply embedded (AV ~ 24 mag) region of less than 120 pc in size. There is also a continuum-emitting dust component. If associated with the putative AGN in IC 694, we estimate that it would be approximately five times less luminous than the AGN in NGC 3690. The presence of dual AGN activity makes Arp 299 a good example to study such phenomena in the early coalescence phase of interacting galaxies.