diff --git a/clippy_lints/src/eta_reduction.rs b/clippy_lints/src/eta_reduction.rs
index 63abe47b4428..60e2ab8b9b9f 100644
--- a/clippy_lints/src/eta_reduction.rs
+++ b/clippy_lints/src/eta_reduction.rs
@@ -1,4 +1,5 @@
 use if_chain::if_chain;
+use matches::matches;
 use rustc::hir::*;
 use rustc::lint::{in_external_macro, LateContext, LateLintPass, LintArray, LintContext, LintPass};
 use rustc::ty::{self, Ty};
@@ -65,6 +66,9 @@ fn check_closure(cx: &LateContext<'_, '_>, expr: &Expr) {
             if !(is_adjusted(cx, ex) || args.iter().any(|arg| is_adjusted(cx, arg)));
 
             let fn_ty = cx.tables.expr_ty(caller);
+
+            if matches!(fn_ty.sty, ty::FnDef(_, _) | ty::FnPtr(_) | ty::Closure(_, _));
+
             if !type_is_unsafe_function(cx, fn_ty);
 
             if compare_inputs(&mut iter_input_pats(decl, body), &mut args.into_iter());
diff --git a/tests/ui/eta.fixed b/tests/ui/eta.fixed
index 7e487fe21679..bae9b09c6977 100644
--- a/tests/ui/eta.fixed
+++ b/tests/ui/eta.fixed
@@ -133,3 +133,19 @@ fn divergent(_: u8) -> ! {
 fn generic<T>(_: T) -> u8 {
     0
 }
+
+fn passes_fn_mut(mut x: Box<dyn FnMut()>) {
+    requires_fn_once(|| x());
+}
+fn requires_fn_once<T: FnOnce()>(_: T) {}
+
+fn test_redundant_closure_with_function_pointer() {
+    type FnPtrType = fn(u8);
+    let foo_ptr: FnPtrType = foo;
+    let a = Some(1u8).map(foo_ptr);
+}
+
+fn test_redundant_closure_with_another_closure() {
+    let closure = |a| println!("{}", a);
+    let a = Some(1u8).map(closure);
+}
diff --git a/tests/ui/eta.rs b/tests/ui/eta.rs
index a0d83c438251..a23da73bde75 100644
--- a/tests/ui/eta.rs
+++ b/tests/ui/eta.rs
@@ -133,3 +133,19 @@ fn divergent(_: u8) -> ! {
 fn generic<T>(_: T) -> u8 {
     0
 }
+
+fn passes_fn_mut(mut x: Box<dyn FnMut()>) {
+    requires_fn_once(|| x());
+}
+fn requires_fn_once<T: FnOnce()>(_: T) {}
+
+fn test_redundant_closure_with_function_pointer() {
+    type FnPtrType = fn(u8);
+    let foo_ptr: FnPtrType = foo;
+    let a = Some(1u8).map(|a| foo_ptr(a));
+}
+
+fn test_redundant_closure_with_another_closure() {
+    let closure = |a| println!("{}", a);
+    let a = Some(1u8).map(|a| closure(a));
+}
diff --git a/tests/ui/eta.stderr b/tests/ui/eta.stderr
index 77423694f815..eb55a251bcc9 100644
--- a/tests/ui/eta.stderr
+++ b/tests/ui/eta.stderr
@@ -68,5 +68,17 @@ error: redundant closure found
 LL |     let e: std::vec::Vec<char> = vec!['a', 'b', 'c'].iter().map(|c| c.to_ascii_uppercase()).collect();
    |                                                                 ^^^^^^^^^^^^^^^^^^^^^^^^^^ help: remove closure as shown: `char::to_ascii_uppercase`
 
-error: aborting due to 11 previous errors
+error: redundant closure found
+  --> $DIR/eta.rs:145:27
+   |
+LL |     let a = Some(1u8).map(|a| foo_ptr(a));
+   |                           ^^^^^^^^^^^^^^ help: remove closure as shown: `foo_ptr`
+
+error: redundant closure found
+  --> $DIR/eta.rs:150:27
+   |
+LL |     let a = Some(1u8).map(|a| closure(a));
+   |                           ^^^^^^^^^^^^^^ help: remove closure as shown: `closure`
+
+error: aborting due to 13 previous errors