  Extragalactic globular clusters (EGCs) are an abundant and powerful tracer of
galaxy dynamics and formation, and their own formation and evolution is also a
matter of extensive debate. The compact nature of globular clusters means that
they are hard to spatially resolve and thus study outside the Local Group. In
this work we have examined how well EGCs will be detectable in images from the
Euclid telescope, using both simulated pre-launch images and the first
early-release observations of the Fornax galaxy cluster. The Euclid Wide Survey
will provide high-spatial resolution VIS imaging in the broad IE band as well
as near-infrared photometry (YE, JE, and HE). We estimate that the galaxies
within 100 Mpc in the footprint of the Euclid survey host around 830 000 EGCs
of which about 350 000 are within the survey's detection limits. For about half
of these EGCs, three infrared colours will be available as well. For any galaxy
within 50Mpc the brighter half of its GC luminosity function will be detectable
by the Euclid Wide Survey. The detectability of EGCs is mainly driven by the
residual surface brightness of their host galaxy. We find that an automated
machine-learning EGC-classification method based on real Euclid data of the
Fornax galaxy cluster provides an efficient method to generate high purity and
high completeness GC candidate catalogues. We confirm that EGCs are spatially
resolved compared to pure point sources in VIS images of Fornax. Our analysis
of both simulated and first on-sky data show that Euclid will increase the
number of GCs accessible with high-resolution imaging substantially compared to
previous surveys, and will permit the study of GCs in the outskirts of their
hosts. Euclid is unique in enabling systematic studies of EGCs in a spatially
unbiased and homogeneous manner and is primed to improve our understanding of
many understudied aspects of GC astrophysics.
