We examine the solvent-mediated interaction between two neutral colloidal particles due to preferential adsorption in a near-critical binary mixture. We take into account the renormalization effect due to the critical fluctuations using the recent local functional theory $[$J. Chem. Phys. {\bf 136}, 114704 (2012)$]$. We calculate the free energy and the force between two colloidal particles as functions of the temperature $T$, the composition far from the colloidal particles $c_\infty$, and the colloid separation $\ell$. The interaction is much enhanced when the component favored by the colloid surfaces is poor in the reservoir. For such off-critical compositions, we find a surface of a first-order bridging transition $\ell= \ell_{\rm cx}(T,c_\infty)$ in the $T$-$c_\infty$-$\ell$ space in a universal, scaled form, across which a discontinuous change occurs between separated and bridged states. This surface starts from the bulk coexistence surface (CX) and ends at a bridging critical line $\ell= \ell_{c}(T)$. On approaching the critical line, the discontinuity vanishes and the derivatives of the force with respect to $T$ and $\ell$ both diverge. Furthermore, bridged states continuously change into separated states if $c_\infty$ (or $T$) is varied from a value on CX to value far from CX with $\ell$ kept smaller than $\ell_c(T)$.

15 pages, 17 figures