diff --git a/module/icp/Makefile.in b/module/icp/Makefile.in
index 54b5df1a19..ce84999ad1 100644
--- a/module/icp/Makefile.in
+++ b/module/icp/Makefile.in
@@ -60,12 +60,9 @@ $(MODULE)-$(CONFIG_X86) += algs/modes/gcm_pclmulqdq.o
 $(MODULE)-$(CONFIG_X86) += algs/aes/aes_impl_aesni.o
 $(MODULE)-$(CONFIG_X86) += algs/aes/aes_impl_x86-64.o
 
-# Suppress objtool "can't find jump dest instruction at" warnings.  They
-# are caused by the constants which are defined in the text section of the
-# assembly file using .byte instructions (e.g. bswap_mask).  The objtool
-# utility tries to interpret them as opcodes and obviously fails doing so.
+# Suppress objtool "return with modified stack frame" warnings.
 OBJECT_FILES_NON_STANDARD_aesni-gcm-x86_64.o := y
-OBJECT_FILES_NON_STANDARD_ghash-x86_64.o := y
+
 # Suppress objtool "unsupported stack pointer realignment" warnings. We are
 # not using a DRAP register while aligning the stack to a 64 byte boundary.
 # See #6950 for the reasoning.
diff --git a/module/icp/asm-x86_64/modes/aesni-gcm-x86_64.S b/module/icp/asm-x86_64/modes/aesni-gcm-x86_64.S
index 7414b3540f..6da43ee005 100644
--- a/module/icp/asm-x86_64/modes/aesni-gcm-x86_64.S
+++ b/module/icp/asm-x86_64/modes/aesni-gcm-x86_64.S
@@ -1242,6 +1242,7 @@ atomic_toggle_boolean_nv:
 	RET
 .size	atomic_toggle_boolean_nv,.-atomic_toggle_boolean_nv
 
+.pushsection .rodata
 .align	64
 .Lbswap_mask:
 .byte	15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0
@@ -1255,6 +1256,7 @@ atomic_toggle_boolean_nv:
 .byte	1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
 .byte	65,69,83,45,78,73,32,71,67,77,32,109,111,100,117,108,101,32,102,111,114,32,120,56,54,95,54,52,44,32,67,82,89,80,84,79,71,65,77,83,32,98,121,32,60,97,112,112,114,111,64,111,112,101,110,115,115,108,46,111,114,103,62,0
 .align	64
+.popsection
 
 /* Mark the stack non-executable. */
 #if defined(__linux__) && defined(__ELF__)
diff --git a/module/icp/asm-x86_64/modes/ghash-x86_64.S b/module/icp/asm-x86_64/modes/ghash-x86_64.S
index 77a3ce1859..d7cdaeb368 100644
--- a/module/icp/asm-x86_64/modes/ghash-x86_64.S
+++ b/module/icp/asm-x86_64/modes/ghash-x86_64.S
@@ -655,6 +655,8 @@ gcm_ghash_avx:
 	RET
 .cfi_endproc
 .size	gcm_ghash_avx,.-gcm_ghash_avx
+
+.pushsection .rodata
 .align	64
 .Lbswap_mask:
 .byte	15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0
@@ -708,6 +710,7 @@ gcm_ghash_avx:
 
 .byte	71,72,65,83,72,32,102,111,114,32,120,56,54,95,54,52,44,32,67,82,89,80,84,79,71,65,77,83,32,98,121,32,60,97,112,112,114,111,64,111,112,101,110,115,115,108,46,111,114,103,62,0
 .align	64
+.popsection
 
 /* Mark the stack non-executable. */
 #if defined(__linux__) && defined(__ELF__)