We derived galaxy colour selections from Euclid and ground-based
photometry, aiming to accurately define background galaxy samples in
cluster weak-lensing analyses. Given any set of photometric bands, we
developed a method for the calibration of optimal galaxy colour
selections that maximises the selection completeness, given a threshold
on purity. We calibrated galaxy selections using simulated ground-based
$griz$ and Euclid $Y_{\rm E}J_{\rm E}H_{\rm E}$ photometry. Both
selections produce a purity higher than 97%. The $griz$ selection
completeness ranges from 30% to 84% in the lens redshift range $z_{\rm
l}\in[0.2,0.8]$. With the full $grizY_{\rm E}J_{\rm E}H_{\rm E}$
selection, the completeness improves by up to $25$ percentage points,
and the $z_{\rm l}$ range extends up to $z_{\rm l}=1.5$. The calibrated
colour selections are stable to changes in the sample limiting
magnitudes and redshift, and the selection based on $griz$ bands
provides excellent results on real external datasets. The $griz$
selection is also purer at high redshift and more complete at low
redshift compared to colour selections found in the literature. We find
excellent agreement in terms of purity and completeness between the
analysis of an independent, simulated Euclid galaxy catalogue and our
calibration sample, except for galaxies at high redshifts, for which we
obtain up to 50 percent points higher completeness. The combination of
colour and photo-$z$ selections applied to simulated Euclid data yields
up to 95% completeness, while the purity decreases down to 92% at high
$z_{\rm l}$. We show that the calibrated colour selections provide
robust results even when observations from a single band are missing
from the ground-based data. Finally, we show that colour selections do
not disrupt the shear calibration for stage III surveys.