We have completed a survey down to R = 15 mag of the stars within a circle of 4 arcmin radius around the nominal center of the remnant of SN 1006, one of the three historical Type Ia supernovae (the other two being SN 1572 and SN 1604), in search of a possible surviving binary companion of the white dwarf whose explosion gave rise to the supernova. The stellar parameters (effective temperature, surface gravity, and metallicity), as well as the radial velocities of all the stars, have been measured from spectra obtained with the UVES spectrograph at the VLT, and from the former and the available photometry, distances have been determined. Chemical abundances of the Fe-peak elements Cr, Mn, Co, and Ni have also been measured to check for possible contamination of the stellar surface by the supernova ejecta. The limiting magnitude of the survey would allow us to find stellar companions of the red-giant type, subgiant stars, and main-sequence stars down to F5-6. Unlike in SN 1572, where a subgiant of type G0-1 has been proposed as the companion of SN 1572, for SN 1006 we can discard the possibility that SN 1006 had a red giant or subgiant companion.