  Part of the intrinsic structure of singular integrals in the Bessel setting
is captured by Muckenhoupt-type weights. Anderson--Kerman showed that the
Bessel Riesz transform is bounded on weighted $L^p_w$ if and only if $w$ is in
the class $A_{p,\lambda}$. We introduce a new class of Muckenhoupt-type weights
$\widetilde A_{p,\lambda}$ in the Bessel setting, which is different from
$A_{p,\lambda}$ but characterizes the weighted boundedness for the
Hardy--Littlewood maximal operators. We also establish the weighted $L^p$
boundedness and compactness, as well as the endpoint weak type boundedness of
Riesz commutators. The quantitative weighted bound is also established.
