  We explore the sensitivity to new physics (NP) in the associated production
of top-quarks with leptons $pp \to t \bar t \ell^+ \ell^-$, which leads to the
multi-leptons signals $pp \to n \ell + {\tt jets} + \not\!\! E_T$, where $n =
2,3,4$. The NP is parameterized via 4-Fermi effective $t\bar{t} \ell^+ \ell^-$
contact interactions of various types, which are generated by multi-TeV heavy
scalar, vector or tensor exchanges in $t \bar t \to \ell^+ \ell^-$; we focus on
the case of $\ell=e,\mu$. We match the 4-Fermi $t t \ell \ell$ terms to the
SMEFT operators and also give examples of specific underlying heavy physics
that can generate such terms. Analysis of the SM signals and corresponding
backgrounds shows that the di-lepton and tri-lepton channels are much better
probes of the effective $t\bar{t} \ell^+ \ell^-$ 4-Fermi terms than the
four-lepton one at the 13 TeV LHC. Therefore, the best sensitivity is obtained
in the di- and tri-lepton channels, for which the dominant background $pp \to t
\bar t$ and $pp \to WZ$, respectively, can be essentially eliminated after
applying the $2\ell$ and $ 3 \ell$ selections and a sufficiently high invariant
mass selection for the opposite sign same flavor (OSSF) lepton-pair. We explore
two cases: lepton flavor universal (LFU) NP where the $t t e e$ and $t t \mu
\mu$ contact interactions are of same size and LFU violating (LFUV) NP, where
the scale of the $t t \mu \mu$ terms is assumed to be much lower. We show that
in both cases it is possible to obtain new 95\% CL bounds on the scale of the
$t t \ell \ell$ contact interactions at the level $\Lambda \gtrsim 2-3$ TeV,
which are considerably tighter than the current bounds on these 4-Fermi terms.
