  We study Flipped $SU(5)\times U(1)$ Grand Unified Theories (GUTs) with
$\Gamma_3\simeq A_4$ modular symmetry. We propose two models with different
modular weights assignments, where the fermion mass hierarchy can arise from
weighton fields. In order to relax the constraint on the Dirac neutrino Yukawa
matrix we appeal to mechanisms which allow incomplete GUT representations,
allowing a good fit to quark and charged lepton masses and quark mixing for a
single modulus field $\tau$, with the neutrino masses and lepton mixing well
determined by the type I seesaw mechanism, at the expense of some tuning. We
also discuss the double seesaw possibility allowed by the extra singlets
generically predicted in such string inspired theories.
