diff --git a/src/librustc_mir/transform/const_prop.rs b/src/librustc_mir/transform/const_prop.rs
index 1d5a643484a73..35dbf2f7eaa42 100644
--- a/src/librustc_mir/transform/const_prop.rs
+++ b/src/librustc_mir/transform/const_prop.rs
@@ -74,6 +74,31 @@ impl<'tcx> MirPass<'tcx> for ConstProp {
             return;
         }
 
+        // Check if it's even possible to satisfy the 'where' clauses
+        // for this item.
+        // This branch will never be taken for any normal function.
+        // However, it's possible to `#!feature(trivial_bounds)]` to write
+        // a function with impossible to satisfy clauses, e.g.:
+        // `fn foo() where String: Copy {}`
+        //
+        // We don't usually need to worry about this kind of case,
+        // since we would get a compilation error if the user tried
+        // to call it. However, since we can do const propagation
+        // even without any calls to the function, we need to make
+        // sure that it even makes sense to try to evaluate the body.
+        // If there are unsatisfiable where clauses, then all bets are
+        // off, and we just give up.
+        if !tcx.substitute_normalize_and_test_predicates((
+            source.def_id(),
+            InternalSubsts::identity_for_item(tcx, source.def_id()),
+        )) {
+            trace!(
+                "ConstProp skipped for item with unsatisfiable predicates: {:?}",
+                source.def_id()
+            );
+            return;
+        }
+
         trace!("ConstProp starting for {:?}", source.def_id());
 
         let dummy_body = &Body::new(
diff --git a/src/test/ui/consts/issue-67696-const-prop-ice.rs b/src/test/ui/consts/issue-67696-const-prop-ice.rs
new file mode 100644
index 0000000000000..ad52608b3f46d
--- /dev/null
+++ b/src/test/ui/consts/issue-67696-const-prop-ice.rs
@@ -0,0 +1,20 @@
+// check-pass
+// compile-flags: --emit=mir,link
+// Checks that we don't ICE due to attempting to run const prop
+// on a function with unsatisifable 'where' clauses
+
+#![allow(unused)]
+
+trait A {
+    fn foo(&self) -> Self where Self: Copy;
+}
+
+impl A for [fn(&())] {
+    fn foo(&self) -> Self where Self: Copy { *(&[] as &[_]) }
+}
+
+impl A for i32 {
+    fn foo(&self) -> Self { 3 }
+}
+
+fn main() {}
diff --git a/src/test/ui/trivial-bounds/trivial-bounds-inconsistent-associated-functions.rs b/src/test/ui/trivial-bounds/trivial-bounds-inconsistent-associated-functions.rs
index 6450ddd1b67fc..69eee66e64d89 100644
--- a/src/test/ui/trivial-bounds/trivial-bounds-inconsistent-associated-functions.rs
+++ b/src/test/ui/trivial-bounds/trivial-bounds-inconsistent-associated-functions.rs
@@ -1,4 +1,8 @@
-// run-pass
+// check-pass
+// compile-flags: --emit=mir,link
+// Force mir to be emitted, to ensure that const
+// propagation doesn't ICE on a function
+// with an 'impossible' body. See issue #67696
 // Inconsistent bounds with trait implementations
 
 #![feature(trivial_bounds)]