A primary goal of the SCUBA-2 Web survey is to perform tomography of the early inter-galactic medium by studying systems containing some of the brightest quasi-stellar objects (QSOs; 2.5<z<3.0) and nearby submillimetre galaxies. As a first step, this paper aims to characterize the galaxies that host the QSOs. To achieve this, a sample of 13 hyper-luminous (L_AGN>10^14 L_odot) QSOs with previous submillimetre continuum detections were followed up with CO(3-2) observations using the NOEMA interferometer. All but two of the QSOs are detected in CO(3-2); for one non-detection, our observations show a tentative 2sigma line at the expected position and redshift, and for the other non-detection we find only continuum flux density an order of magnitude brighter than the other sources. In three of the fields, a companion potentially suitable for tomography is detected in CO line emission within 25 arcsec of the QSO. We derive gas masses, dynamical masses and far-infrared luminosities, and show that the QSOs in our sample have similar properties as compared to less luminous QSOs and SMGs in the literature, despite the fact that their black-hole masses (which are proportional to L_AGN) are 1-2 orders of magnitude larger. We discuss two interpretations of these observations: this is due to selection effects, such as preferential face-on viewing angles and picking out objects in the tail ends of the scatter in host-galaxy mass and black-hole mass relationships; or the black hole masses have been overestimated because the accretion rates are super-Eddington.
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