diff --git a/core/modules/update/templates/update-version.html.twig b/core/modules/update/templates/update-version.html.twig
index dd212b94af..0cb6df4f60 100644
--- a/core/modules/update/templates/update-version.html.twig
+++ b/core/modules/update/templates/update-version.html.twig
@@ -26,7 +26,7 @@
     </div>
     <div class="layout-column layout-column--half">
       <ul class="project-update__version-links">
-        {% if version.core_compatible %}
+        {% if version.show_download_link %}
           <li class="project-update__download-link">
             <a href="{{ version.download_link }}">{{ 'Download'|t }}</a>
           </li>
diff --git a/core/modules/update/update.report.inc b/core/modules/update/update.report.inc
index 92da6780c5..9dadc58dc9 100644
--- a/core/modules/update/update.report.inc
+++ b/core/modules/update/update.report.inc
@@ -114,6 +114,14 @@ function template_preprocess_update_project_status(&$variables) {
   // Storing by reference because we are sorting the project values.
   $project = &$variables['project'];
 
+  $set_download_link_flag = function (&$version) use ($project) {
+    if ($project['name'] === 'drupal' || !empty($version['core_compatible'])) {
+      $version['show_download_link'] = TRUE;
+      return;
+    }
+    $version['show_download_link'] = FALSE;
+  };
+
   // Set the project title and URL.
   $variables['title'] = (isset($project['title'])) ? $project['title'] : $project['name'];
   $variables['url'] = (isset($project['link'])) ? Url::fromUri($project['link'])->toString() : NULL;
@@ -157,6 +165,7 @@ function template_preprocess_update_project_status(&$variables) {
         ) {
           $version_class[] = 'project-update__version--recommended-strong';
         }
+        $set_download_link_flag($project['releases'][$project['recommended']]);
         $versions_inner[] = [
           '#theme' => 'update_version',
           '#version' => $project['releases'][$project['recommended']],
@@ -169,6 +178,7 @@ function template_preprocess_update_project_status(&$variables) {
       if (!empty($project['security updates'])) {
         $security_class[] = 'version-security';
         foreach ($project['security updates'] as $security_update) {
+          $set_download_link_flag($security_update);
           $versions_inner[] = [
             '#theme' => 'update_version',
             '#version' => $security_update,
@@ -180,6 +190,7 @@ function template_preprocess_update_project_status(&$variables) {
     }
 
     if ($project['recommended'] !== $project['latest_version']) {
+      $set_download_link_flag($project['releases'][$project['latest_version']]);
       $versions_inner[] = [
         '#theme' => 'update_version',
         '#version' => $project['releases'][$project['latest_version']],
@@ -191,6 +202,7 @@ function template_preprocess_update_project_status(&$variables) {
       && $project['status'] != UpdateManagerInterface::CURRENT
       && isset($project['dev_version'])
       && $project['recommended'] !== $project['dev_version']) {
+      $set_download_link_flag($project['releases'][$project['dev_version']]);
       $versions_inner[] = [
         '#theme' => 'update_version',
         '#version' => $project['releases'][$project['dev_version']],
@@ -202,6 +214,7 @@ function template_preprocess_update_project_status(&$variables) {
 
   if (isset($project['also'])) {
     foreach ($project['also'] as $also) {
+      $set_download_link_flag($project['releases'][$also]);
       $versions_inner[] = [
         '#theme' => 'update_version',
         '#version' => $project['releases'][$also],
diff --git a/core/themes/claro/templates/admin/update-version.html.twig b/core/themes/claro/templates/admin/update-version.html.twig
index df82642ac0..0704aec2fb 100644
--- a/core/themes/claro/templates/admin/update-version.html.twig
+++ b/core/themes/claro/templates/admin/update-version.html.twig
@@ -24,7 +24,7 @@
     </div>
     <div class="layout-column layout-column--half">
       <ul class="project-update__version-links">
-        {% if version.core_compatible %}
+        {% if version.show_download_link %}
           <li class="project-update__download-link">
             <a href="{{ version.download_link }}">{{ 'Download'|t }}</a>
           </li>
diff --git a/core/themes/stable/templates/admin/update-version.html.twig b/core/themes/stable/templates/admin/update-version.html.twig
index 6b135119a3..6e50069dd0 100644
--- a/core/themes/stable/templates/admin/update-version.html.twig
+++ b/core/themes/stable/templates/admin/update-version.html.twig
@@ -24,7 +24,7 @@
     </div>
     <div class="layout-column layout-column--half">
       <ul class="project-update__version-links">
-        {% if version.core_compatible %}
+        {% if version.show_download_link %}
           <li class="project-update__download-link">
             <a href="{{ version.download_link }}">{{ 'Download'|t }}</a>
           </li>