Context. We report on the first of a series of studies on coronal bright points which investigate the physical mechanism that generates these phenomena. Aims: The aim of this paper is to understand the magnetic-field structure that hosts the bright points. Methods: We use longitudinal magnetograms taken by the Solar Optical Telescope with the Narrowband Filter Imager. For a single case, magnetograms from the Helioseismic and Magnetic Imager were added to the analysis. The longitudinal magnetic field component is used to derive the potential magnetic fields of the large regions around the bright points. A magneto-static field extrapolation method is tested to verify the accuracy of the potential field modelling. The three dimensional magnetic fields are investigated for the presence of magnetic null points and their influence on the local magnetic domain. Results: In nine out of ten cases the bright point resides in areas where the coronal magnetic field contains an opposite polarity intrusion defining a magnetic null point above it. We find that X-ray bright points reside, in these nine cases, in a limited part of the projected fan-dome area, either fully inside the dome or expanding over a limited area below which typically a dominant flux concentration resides. The tenth bright point is located in a bipolar loop system without an overlying null point. Conclusions: All bright points in coronal holes and two out of three bright points in quiet Sun regions are seen to reside in regions containing a magnetic null point. An as yet unidentified process(es) generates the brigh points in specific regions of the fan-dome structure. The movies are available at http://www.aanda.org