White dwarf spectroscopy shows that nearly half of white dwarf atmospheres contain metals that must have been accreted from planetary material that survived the red giant phases of stellar evolution. We can use metal pollution in white dwarf atmospheres as flags, signalling recent accretion, in order to prioritize an efficient sample of white dwarfs to search for transiting material. We present a search for planetesimals orbiting six nearby white dwarfs with the CHaracterising ExOPlanet Satellite (CHEOPS). The targets are relatively faint for CHEOPS, 11 mag<G<12.8 mag. We used aperture photometry data products from the CHEOPS mission as well as custom point-spread function photometry to search for periodic variations in flux due to transiting planetesimals. We detect no significant variations in flux that cannot be attributed to spacecraft systematics, despite reaching a photometric precision of < 2 ppt in 60 s exposures on each target. We simulate observations to show that the small survey is sensitive primarily to Moon-sized transiting objects with periods between 3 h<P<10 h, with radii of R<greater than or similar to>1000 km.