Sagittarius A* (Sgr A*) at the center of the Milky Way is a black hole accreting at extremely sub-Eddington rates. Measurements of its emission in the infrared and X-ray domains are difficult due to its faintness and high variability. The Galactic center was observed at 8.6 micron in order to detect a mid-infrared (MIR) counterpart to Sgr A*, parallel to NIR observations. The goal was to set constraints on possible emission mechanisms. Imaging data were acquired with the adaptive-optics assisted NIR instrument NACO and the MIR instrument VISIR at the ESO VLT. We present MIR imaging data of an unprecedented quality in terms of spatial resolution and sensitivity. An extended ridge of emission is found to be present in the immediate vicinity of Sgr A* thereby rendering any detection of a point source difficult. No MIR point source related to Sgr A* was detected during the observations. We derive a tight upper limit of 22+/-14 mJy (dereddened) on any possible point source present during the observations in the night of 4/5 June 2006. The absence of a flare in simultaneous observations at 2.2 micron and the low limits on any possible variability in the MIR strongly suggest that Sgr A* was in a quasi-quiescent state during this night. During the night from 5 to 6 June 2006, Sgr A* was found to be variable on a low level at 3.8 micron . No point source at 8.6 micron was detected during the simultaneous MIR observations. Due to the poorer atmospheric conditions, a higher upper limit of 60+/-30 mJy was found for Sgr A* at 8.6 micron during the second night. The observations are consistent with theoretical predictions. If the published models are correct, the observations demonstrate successfully that a 8.6 micron counterpart of Sgr A* can be easily detected in its flaring state. Spectral indices derived from simultaneous observations of flaring emission from Sgr A* at NIR and MIR wavelengths will enable us to distinguish between different kinds of flare models.