  We carry out an in-depth analysis of the capability of the upcoming
space-based gravitational wave mission eLISA in addressing the Hubble tension,
with a primary focus on observations at intermediate redshifts ($3<z<8$). We
consider six different parametrizations representing different classes of
cosmological models, which we constrain using the latest datasets of cosmic
microwave background (CMB), baryon acoustic oscillations (BAO), and type Ia
supernovae (SNIa) observations, in order to find out the up-to-date tensions
with direct measurement data. Subsequently, these constraints are used as
fiducials to construct mock catalogs for eLISA. We then employ Fisher analysis
to forecast the future performance of each model in the context of eLISA. We
further implement traditional Markov Chain Monte Carlo (MCMC) to estimate the
parameters from the simulated catalogs. Finally, we utilize Gaussian Processes
(GP), a machine learning algorithm, for reconstructing the Hubble parameter
directly from simulated data. Based on our analysis, we present a thorough
comparison of the three methods as forecasting tools. Our Fisher analysis
confirms that eLISA would constrain the Hubble constant ($H_0$) at the
sub-percent level. MCMC/GP results predict reduced tensions for
models/fiducials which are currently harder to reconcile with direct
measurements of $H_0$, whereas no significant change occurs for
models/fiducials at lesser tensions with the latter. This feature warrants
further investigation in this direction.
