  The perspectives of detecting the general relativistic gravitomagnetic
Lense-Thirring effect on the orbits of the Galilean moons of Jupiter induced by
the angular momentum ${\boldsymbol{S}}$ of the latter are preliminarily
investigated. Numerical integrations over one century show that the expected
gravitomagnetic signatures of the directly observable right ascension $\alpha$
and declination $\delta$ of the satellites are as large as tens of arcseconds
for Io, while for Callisto they drop to the $\simeq 0.2\,\mathrm{arcseconds}$
level. Major competing effects due to the mismodeling in the zonal multipoles
$J_\ell,\,\ell=2,\,3,\,4,\,\ldots$ of the Jovian non-spherically symmetric
gravity field and in the Jupiter's spin axis ${\boldsymbol{\hat{k}}}$ should
have a limited impact, especially in view of the future improvements in
determining such parameters expected after the completion of the ongoing Juno
mission in the next few years. On the other hand, the masses of the satellites,
responsible of their mutual $N-$body perturbations, should be known better than
now. Such a task should be accomplished with the future JUICE and Clipper
missions to the Jovian system. Present-day accuracy in knowing the orbits of
the Jovian Galilean satellites is of the order of 10 milliarcseconds, to be
likely further improved thanks to the ongoing re-reduction of old photographic
plates. This suggests that, in the next future, the Lense-Thirring effect in
the main Jovian system of moons might be detectable with dedicated data
reductions in which the gravitomagnetic field is explicitly modeled and
solved-for.
