  We study the classical problem of Kostant for Whittaker modules over Lie
algebras and Lie superalgebras. We give a sufficient condition for a positive
answer to Kostant's problem for the standard Whittaker modules over reductive
Lie algebras. Under the same condition, the positivity of the answer for simple
Whittaker modules is reduced to that for simple highest weight modules. We
develop several reduction results to reduce the Kostant's problem for standard
and simple Whittaker modules over a type I Lie superalgebra to that for the
corresponding Whittaker modules over the even part of this Lie superalgebra.
