  The rate at which the universe is expanding today is a fundamental parameter
in cosmology which governs our understanding of structure formation and dark
energy. However, current measurements of the Hubble constant, $H_0$, show a
significant tension ($\sim 4-6\sigma$) between early- and late-Universe
observations. There are ongoing efforts to check the diverse observational
results and also to investigate possible theoretical ways to resolve the
tension~-- which could point to radical extensions of the standard model. Here
we demonstrate the potential of next-generation spectroscopic galaxy surveys to
shed light on the Hubble constant tension. Surveys such as those with Euclid
and the Square Kilometre Array (SKA) are expected to reach sub-percent
precision on Baryon Acoustic Oscillation (BAO) measurements of the Hubble
parameter, with a combined redshift coverage of $0.1<z<3$. This wide redshift
range, together with the high precision and low level of systematics in BAO
measurements, mean that these surveys will provide independent and tight
constraints on $H(z)$. These $H(z)$ measurements can be extrapolated to $z = 0$
to provide constraints on $H_0$ using a non-parametric regression. To this end
we deploy Gaussian processes and we find that Euclid-like surveys can reach
$\sim$3\% precision on $H_0$, with SKA-like intensity mapping surveys reaching
$\sim$2\%. When we combine the low-redshift SKA-like Band 2 survey with either
its high-redshift Band 1 counterpart, or with the non-overlapping Euclid-like
survey, the precision is predicted to be close to 1\% with 40 $H(z)$ data
points. This would be sufficient to rule out the current early- or
late-Universe measurements at a $\sim$5$\sigma$ level.
