Using the method of localization, the authors obtain the permutation formula of singular integrals with Bochner-Martinelli kernel for a relative compact domain with C⁽¹⁾ smooth boundary on a Stein manifold. As an application the authors discuss the regularization problem for linear singular integral equations with Bochner--Martinelli kernel and variable coefficients; using permutation formula, the singular integral equation can be reduced to a fredholm equation.