  Recently, the bulk nickelate La$_3$Ni$_2$O$_7$ is reported to show signature
of high-temperature superconductivity under high pressure above $14$GPa [H. Sun
et al., Nature 621, 493 (2023)]. We analyze the pairing mechanism and pairing
symmetry in a bilayer Hubbard model with two orbitals in the $E_g$ multiplet.
In the weak to moderate interaction regime, our functional renormalization
group (FRG) calculations yield $S_\pm$-wave Cooper pairing triggered by leading
spin fluctuations. The gap function changes sign across the Fermi pockets, and
in real space the pairing is dominated by intra-unitcell intra-orbital
components with antiphase between the onsite ones. In the strong coupling
limit, we develop a low-energy effective theory in terms of atomic one- and
two-electron states in the $E_g$ multiplet. The variational treatment of the
effective theory produces results consistent with the FRG ones, suggesting the
robustness of such a pairing function. The driving force for superconductivity
in the strong coupling limit can be attributed to the local pair-hopping term
and the spin-exchange on vertical bonds. We also discuss a possible scenario
for the weak insulating behavior under low pressures in terms of the tendency
toward the formation of charge order in the strong coupling limit.
