The concept of an analogue of magnetic charge is introduced into the Yang-Mills field by reference to local singularities of the classical field. On the basis of the gauge-invariant and path-dependent formalism, the one-valuedness of the path-dependent quantities for a scalar field interacting with the Yang-Mills field imposes a quantum condition on the coupling constant. When the Yang-Mills field is analytic in spacetime, this property leads to consistency conditions analogous to the homogeneous Maxwell equations and to Bianchi identities in the theory of the general relativity. A simple example with a non-trivial (non-zero quantum number) quantum condition is presented in a way compatible with the framework of non-relativistic quantum mechanics, where the Yang-Mills field works on the scalar field as an external field. Additional remarks on more general gauge fields are appended.