  We give a construction of $p$-adic Asai $L$-functions for cohomological
cuspidal automorphic representations of ${\rm GL}_2$ over CM fields. If the
base field is imaginary quadratic, Loeffler-Williams recently constructed the
$p$-adic Asai $L$-functions. We generalize their construction to the case that
the base fields are general CM fields.
