  We describe twisted configurations of spinor field on the Schwarzschild and
Reissner-Nordstr\"om black holes that arise due to existence of the twisted
spinor bundles over the standard black hole topology. From a physical point of
view the appearance of spinor twisted configurations is linked with the natural
presence of Dirac monopoles that play the role of connections in the complex
line bundles corresponding to the twisted spinor bundles. Possible application
to the Hawking radiation is also outlined.
