The goal of this paper is to establish the boundedness of a bilinear strongly Calderón-Zygmund singular integral operator \({\widetilde{T}}\) whose kernel satisfies weaker smooth conditions and its commutator \({\widetilde{T}}_{b_{1},b_{2}}\) formed by \(b_{1}, b_{2}\in \textrm{BMO}(\mu )\) and the \({\widetilde{T}}\) on Lebesgue spaces \(L^{p}(\mu )\) , Morrey spaces \({\mathcal {M}}^{p,\kappa }(\mu )\) and generalized Morrey spaces \({\mathcal {L}}^{p,\varphi }(\mu )\) over RD-spaces. Via establishing the sharp maximal estimates for the \({\widetilde{T}}\) and \({\widetilde{T}}_{b_{1},b_{2}}\) , the author proves that the \({\widetilde{T}}\) and \({\widetilde{T}}_{b_{1},b_{2}}\) are respectively bounded from product of spaces \(L^{p_{1}}(\mu )\times L^{p_{2}}(\mu )\) into spaces \(L^{p}(\mu )\) , where \(\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\) for \(1<\) \(p'_{0}<p_{1}, p_{2}<\infty \) . Furthermore, via using some known results, the author shows that the \({\widetilde{T}}\) and \({\widetilde{T}}_{b_{1},b_{2}}\) are bounded from spaces \({\mathcal {M}}^{p_{1},\kappa }(\mu )\times {\mathcal {M}}^{p_{2},\kappa }(\mu )\) into spaces \({\mathcal {M}}^{p,\kappa }(\mu )\) , and they are also bounded from spaces \({\mathcal {L}}^{p_{1},\varphi _{1}}(\mu )\times {\mathcal {L}}^{p_{2},\varphi _{2}}(\mu )\) into spaces \({\mathcal {L}}^{p,\varphi }(\mu )\) , where \(0<\kappa <1\) , \(\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\) for \(p'_{0}<p_{1}, p_{2}<\infty \) , and the Lebesgue measurable functions \(\varphi _{1}, \varphi _{2}, \varphi \) belong to the class \({\mathbb {W}}_{\tau }\) and satisfy \(\varphi _{1}\varphi _{2}=\varphi \) .
