Using the form factors of the transtions $B\to T$ with $T$ refering to a tensor meson, such as $a_2(1320),f_2(1270),K_2^{*}(1430),D_2^{*}(2460)$ and $D_{2s}^{*}(2573)$, within the covariant light-front quark model (CLFQM), we provide a detailed investigation of the corresponding semi-leptonic decays $B\to T\ell {\nu }_{\ell }$ with $\ell =e,\nu ,\tau$. All the branching ratios of these decays are larger than ${10}^{-5}$, in which the maximum value can reach up to ${10}^{-3}$, indicating promising prospects for experimental observation. Furthermore, we also calculate the longitudinal polarization fractions $f_L$ and forward-backward asymmetries $A_{FB}$ for these considered decays. All the decays $B\to T\ell {\nu }_{\ell }$ are dominated by the longitudinal polarization, where the polarization fractions can reach up to $\sim 70\%$ for the decays $B\to T{\ell }^{\prime }{\nu }_{{\ell }^{\prime }}$ with ${\ell }^{\prime }=e,\mu$, those of the decays $B\to T\tau {\nu }_{\tau }$ are a little smaller. The $A_{FB}$ values of the decays $B\to T{\ell }^{\prime }{\nu }_{{\ell }^{\prime }}$ and $B\to T\tau {\nu }_{\tau }$ have opposite signs.