  We study gravitational perturbations on the near-horizon region of extremal
and near-extremal rotating black holes in a general higher-derivative extension
of Einstein gravity. We find a decoupled modified Teukolsky equation that rules
the gravitational perturbations and that separates into an angular and a radial
equation. The angular equation leads to a deformation of the spin-weighted
spheroidal harmonics, while the radial equation takes the same form as in Kerr
except for a modification of the angular separation constants. We provide a
detailed analysis of the corrections to these angular separation constants and
find analytic results for axisymmetric modes as well as in the eikonal limit.
As an application, we reproduce recent results that show that extremal Kerr
black holes in higher-derivative gravity become singular under certain
deformations and extend them by including parity-breaking corrections, which we
show lead to the same effect. Finally, we obtain constraints on the form of the
full modified Teukolsky radial equation by demanding that it has the right
near-horizon limit. These results serve as an stepping stone towards the study
of quasinormal modes of near-extremal black holes in higher-derivative
extensions of GR.
