diff --git a/libgomp/testsuite/libgomp.c++/static-aggr-constructor-destructor-1.C b/libgomp/testsuite/libgomp.c++/static-aggr-constructor-destructor-1.C
index 403a071c0c01c4bf187c85ccc5b249b348345892..b5aafc8cabc2f0ed3fba147aeab89839b71df9e9 100644
--- a/libgomp/testsuite/libgomp.c++/static-aggr-constructor-destructor-1.C
+++ b/libgomp/testsuite/libgomp.c++/static-aggr-constructor-destructor-1.C
@@ -9,9 +9,18 @@
 
 // { dg-final { scan-tree-dump-not "omp_is_initial_device" "optimized" } }
 // { dg-final { scan-tree-dump-not "__omp_target_static_init_and_destruction" "optimized" } }
-// FIXME: should be '-not' not '-times' 1:
-// { dg-final { scan-tree-dump-times "void _GLOBAL__off_I_v1" 1 "optimized" } }
-// { dg-final { scan-tree-dump-times "__attribute__\\(\\(\[^\n\r]*omp declare target nohost" 1 "optimized" } }
+
+// (A) No offloading configured: The symbols aren't present
+//     Caveat: They are present with -foffload=disable - or offloading
+//     configured but none of the optional offload packages/binaries installed.
+//     But the 'offload_target_any' check cannot distinguish those
+// { dg-final { scan-tree-dump-not "void _GLOBAL__off_I_v1" "optimized" { target { ! offload_target_any } } } }
+// { dg-final { scan-tree-dump-not "__attribute__\\(\\(\[^\n\r]*omp declare target nohost" "optimized" { target { ! offload_target_any } } } }
+
+// (B) With offload configured (and compiling for an offload target)
+//     the symbols are present (missed optimization). Hence: FIXME.
+// { dg-final { scan-tree-dump-times "void _GLOBAL__off_I_v1" 1 "optimized" { target offload_target_any } } }
+// { dg-final { scan-tree-dump-times "__attribute__\\(\\(\[^\n\r]*omp declare target nohost" 1 "optimized" { target offload_target_any } } }
 
 // { dg-final { only_for_offload_target amdgcn-amdhsa scan-offload-tree-dump-not "omp_initial_device;" "optimized" { target offload_target_amdgcn } } }
 // { dg-final { only_for_offload_target amdgcn-amdhsa scan-offload-tree-dump "v1\\._x = 5;" "optimized" { target offload_target_amdgcn } } }
diff --git a/libgomp/testsuite/libgomp.c++/static-aggr-constructor-destructor-2.C b/libgomp/testsuite/libgomp.c++/static-aggr-constructor-destructor-2.C
index 6dd4260a522cb9783be0d5d18f42c492344d4121..9652a721bbe273ffabc9c226a56402a98f858452 100644
--- a/libgomp/testsuite/libgomp.c++/static-aggr-constructor-destructor-2.C
+++ b/libgomp/testsuite/libgomp.c++/static-aggr-constructor-destructor-2.C
@@ -9,9 +9,19 @@
 
 // { dg-final { scan-tree-dump-not "omp_is_initial_device" "optimized" } }
 // { dg-final { scan-tree-dump-not "__omp_target_static_init_and_destruction" "optimized" } }
-// FIXME: should be '-not' not '-times' 1:
-// { dg-final { scan-tree-dump-times "void _GLOBAL__off_I_" 1 "optimized" } }
-// { dg-final { scan-tree-dump-times "__attribute__\\(\\(\[^\n\r]*omp declare target nohost" 1 "optimized" } }
+
+// (A) No offloading configured: The symbols aren't present
+//     Caveat: They are present with -foffload=disable - or offloading
+//     configured but none of the optional offload packages/binaries installed.
+//     But the 'offload_target_any' check cannot distinguish those
+// { dg-final { scan-tree-dump-not "void _GLOBAL__off_I_v1" "optimized" { target { ! offload_target_any } } } }
+// { dg-final { scan-tree-dump-not "__attribute__\\(\\(\[^\n\r]*omp declare target nohost" "optimized" { target { ! offload_target_any } } } }
+
+// (B) With offload configured (and compiling for an offload target)
+//     the symbols are present (missed optimization). Hence: FIXME.
+// { dg-final { scan-tree-dump-times "void _GLOBAL__off_I_v1" 1 "optimized" { target offload_target_any } } }
+// { dg-final { scan-tree-dump-times "__attribute__\\(\\(\[^\n\r]*omp declare target nohost" 1 "optimized" { target offload_target_any } } }
+
 
 // { dg-final { only_for_offload_target amdgcn-amdhsa scan-offload-tree-dump-not "omp_initial_device;" "optimized" { target offload_target_amdgcn } } }
 // { dg-final { only_for_offload_target amdgcn-amdhsa scan-offload-tree-dump "v1\\._x = 5;" "optimized" { target offload_target_amdgcn } } }