Measurements of the electron density of populations of extragalactic H II regions in nearby galaxies remain limited, despite the relevance of this quantity for characterizing the porosity of the interstellar medium and the escape of the ionizing radiation. We initiated a project aimed at analysing the root-mean-square electron density ${\langle }n_\mathrm{ e}{\rangle }_{\mathrm{rms}}$, the in situ density ($n_\mathrm{ e}$) and the volume filling factor ($\phi$) of extragalactic H II regions, investigating the dependence of these attributes on nebular and host galaxy properties. We present an image-segmentation methodology for constructing homogeneous H II region catalogues, and apply it to two pilot galaxies: NGC 2403 and NGC 628. We derive ${\langle }n_\mathrm{ e}{\rangle }_{\mathrm{rms}}$ from their H $\alpha$ luminosities and equivalent radii ($R_{\rm eq}$), and obtain $n_\mathrm{ e}$ and $\phi$ for spectroscopic subsamples. While $n_\mathrm{ e}$ is below 300 cm$^{-3}$, ${\langle }n_\mathrm{ e}{\rangle }_{\mathrm{rms}}$ is typically one to two orders of magnitude lower, implying that $\phi$ is in the range ${\sim} 10^{-4}$ to $10^{-1}$. The two galaxies exhibit a similar size─density relation, ${\langle }n_\mathrm{ e}{\rangle }_{\mathrm{rms}}\propto R_{\rm eq}^{-0.3}$, which breaks for $R_{\rm eq}\gtrsim 50$ pc, show at most a weak dependence of ${\langle }n_\mathrm{ e}{\rangle }_{\mathrm{rms}}$ on galactocentric radius for NGC 2403, and no clear dependence of $n_\mathrm{ e}$ or $\phi$ on these parameters. Combining these results with published data, ${\langle }n_\mathrm{ e}{\rangle }_{\mathrm{rms}}$ presents tentative scaling relations with the median H II region size, the fraction of large regions in the parent galaxy, and the star formation rate surface density. These trends, if confirmed, would provide new constraints for massive cluster formation models and important clues for interpreting dependencies observed at high redshift, underscoring the necessity of consistently extending this analysis to larger samples.