  The Milnor class is a generalization of the Milnor number, defined as the
difference (up to sign) of Chern--Schwartz--MacPherson's class and
Fulton--Johnson's canonical Chern class of a local complete intersection
variety in a smooth variety. In this paper we introduce a "motivic"
Grothendieck group $K^{\mathcal Prop}_{\ell.c.i}(\mathcal V/X \to S)$ and
natural transformations from this Grothendieck group to the homology theory. We
capture the Milnor class, more generally Hirzebruch--Milnor class, as a special
value of a distinguished element under these natural transformations. We also
show a Verdier-type Riemann--Roch formula for our motivic Hirzebruch-Milnor
class. We use Fulton--MacPherson's bivariant theory and the motivic Hirzebruch
class.
