  We investigate the transverse energy-energy correlators (TEEC) in the
small-$x$ regime at the upcoming Electron-Ion Collider (EIC). Focusing on the
back-to-back production of electron-hadron pairs in both $ep$ and $eA$
collisions, we establish a factorization theorem given in terms of the hard
function, quark distributions, soft functions, and TEEC jet functions, where
the gluon saturation effect is incorporated. Numerical results for TEEC in both
$ep$ and $eA$ collisions are presented, together with the nuclear modification
factor $R_A$. Our analysis reveals that TEEC observables in deep inelastic
scattering provide a valuable approach for probing gluon saturation phenomena.
Our findings underscore the significance of measuring TEEC at the EIC,
emphasizing its efficacy in advancing our understanding of gluon saturation and
nuclear modifications in high-energy collisions.
