The recent astrometric discovery of the nearby (590pc), massive (33M_{sun}_) dormant black hole candidate Gaia BH3 offers the possibility to resolve angularly the black hole from its companion star using optical interferometry. We aim at detecting emission in the near-infrared K band from the close-in environment of Gaia BH3 caused by accretion. Gaia BH3 (LS II +14 13) has been observed using the GRAVITY instrument with the four 8-m Unit Telescopes of the VLT Interferometer. We searched for the signature of emission of Gaia BH3 in the interferometric observables using the CANDID, PMOIRED and exoGravity tools. With a separation of 18mas, the Gaia BH3 system is well resolvable angularly by GRAVITY. We did not detect emission from the black hole at a contrast level of {Delta}m=6.8mag with respect to the companion star, that is, f_BH_/f_star_<0.2%. This corresponds to an upper limit on the continuum flux density of f_BH_<1.9*10^-16^W/m^2^/micron in the K band. We also did not detect emission from the black hole in the hydrogen BrGamma line. The non-detection of near-infrared emission from the black hole in Gaia BH3 indicates that its accretion of the giant star wind is presently occurring at most at a very low rate. This is consistent with the limit of f_Edd_<4.9*10^-7^ derived previously on the Eddington ratio for an advection-dominated accretion flow. Deeper observations with GRAVITY may be able to detect the black hole as the companion star approaches periastron around year 2030.