  We performed $^{13}$C-NMR experiment and measured spin-lattice relaxation
rate divided by temperature $1/T_{1}T$ near the superconducting (SC) transition
temperature $T_{c}$ in $\kappa$-(BEDT-TTF)$_{2}$Cu[N(CN)$_{2}$]Br ($\kappa$-Br
salt), and $\kappa$-(BEDT-TTF)$_{2}$Cu(NCS)$_{2}$ ($\kappa$-NCS salt). We
observed the reduction of $1/T_{1}T$ starting at the temperature higher than
$T_c$ in $\kappa$-Br salt. Microscopic observation of quasi-particle density of
states in the fluctuating SC state revealed the effects of short-range Cooper
pairs induced in the normal state to the quasi-particle density of states. We
also performed systematic measurements in the fields both parallel and
perpendicular to the conduction plane in $\kappa$-Br and $\kappa$-NCS salts,
and confirmed that the reduction of $1/T_{1}T$ above $T_{c}$ is observed only
in $\kappa$-Br salt regardless of the external field orientation.
