Chandra and XMM-Newton have resolved the 2-8 keV X-ray background (XRB) into point sources. Many of the fainter sources are obscured active galactic nuclei (AGN) with column densities in the range of 10 2 2 - 10 2 3 cm - 2 , some of which have quasar-like luminosities. According to obscuration models, the XRB above 8 keV is dominated by emission from Compton-thick AGN, with column densities exceeding 1.5 X 10 2 4 cm - 2 . Here, we consider whether Compton-thick quasars are detectable by Chandra and XMM-Newton by their direct (i.e. not scattered) X-ray emission. Detectability is optimized if the objects individually have a high luminosity and high redshift, so that the direct emission has a significant flux in the observed band. Using a simple galaxy formation model incorporating accreting black holes, in which quasars build most of their mass in a Compton-thick manner before expelling the obscuring matter, we predict that moderately deep 100-ks Chandra and XMM-Newton exposures may contain a handful of detectable Compton-thick quasars. Deep Ms or more Chandra images should contain 50-100 distant, optically faint, Compton-thick sources. In passing we show that radiation pressure can be as effective in expelling the obscuring gas as quasars winds, and yields a black hole mass proportional to the velocity dispersion of the host bulge to the fourth power.