diff --git a/Library/Homebrew/extend/os/linux/linkage_checker.rb b/Library/Homebrew/extend/os/linux/linkage_checker.rb
index 8e42455b5f..95146eb1f4 100644
--- a/Library/Homebrew/extend/os/linux/linkage_checker.rb
+++ b/Library/Homebrew/extend/os/linux/linkage_checker.rb
@@ -26,12 +26,27 @@ class LinkageChecker
   ].freeze
 
   def display_deprecated_warning(strict: false)
-    return unless @libcrypt_found
-
-    # Steps when removing this entirely:
+    # Steps when moving these to `odisabled`:
+    # - Remove the old library from SYSTEM_LIBRARY_ALLOWLIST above.
+    # - Remove the `disable` and `disable_for_developer` kwargs here.
+    # - Adjust the `broken_library_linkage?` override below to not check for the library.
+    # - Remove the relevant `fail_on_lib*?`.
+    # If there's no more deprecations left:
+    # - Remove the `broken_library_linkage?` override and the generic alias in HOMEBREW_LIBRARY/linkage_checker.rb.
+    #
+    # Steps when removing handling for a library entirely (assuming the steps to `odisabled` has already been done):
+    # - Remove the relevant setting of `@lib*_found` in `check_dylibs` below.
+    # - Remove the `odisabled` line
+    # If there's no library deprecated/disabled handling left:
     # - Remove the `display_` overrides here and the associated generic aliases in HOMEBREW_LIBRARY/linkage_checker.rb
-    # - Remove the setting of `@libcrypt_found` in `check_dylibs` below.
-    odisabled "linkage to libcrypt.so.1", "libcrypt.so.2 in the libxcrypt formula"
+
+    odisabled "linkage to libcrypt.so.1", "libcrypt.so.2 in the libxcrypt formula" if @libcrypt_found
+
+    return unless @libnsl_found
+
+    odeprecated "linkage to libnsl.so.1", "libnsl.so.3 in the libnsl formula",
+                disable:                fail_on_libnsl1?(strict: strict),
+                disable_for_developers: false
   end
 
   def display_normal_output
@@ -45,15 +60,20 @@ class LinkageChecker
   end
 
   def broken_library_linkage?(test: false, strict: false)
-    generic_broken_library_linkage?(test: test, strict: strict)
+    generic_broken_library_linkage?(test: test, strict: strict) || (fail_on_libnsl1?(strict: strict) && @libnsl_found)
   end
 
   private
 
+  def fail_on_libnsl1?(strict:)
+    strict || ENV["HOMEBREW_DISALLOW_LIBNSL1"].present?
+  end
+
   def check_dylibs(rebuild_cache:)
     generic_check_dylibs(rebuild_cache: rebuild_cache)
 
     @libcrypt_found = true if @system_dylibs.any? { |s| File.basename(s) == "libcrypt.so.1" }
+    @libnsl_found = true if @system_dylibs.any? { |s| File.basename(s) == "libnsl.so.1" }
 
     # glibc and gcc are implicit dependencies.
     # No other linkage to system libraries is expected or desired.