diff --git a/lib/ast-utils.js b/lib/ast-utils.js
index 0a97334..51d4bf0 100644
--- a/lib/ast-utils.js
+++ b/lib/ast-utils.js
@@ -7,6 +7,8 @@
 
 "use strict";
 
+const path = require("path");
+
 module.exports = {
   isTypeScriptParserServices(parserServices) {
     // Check properties specific to @typescript-eslint/parser
@@ -39,6 +41,15 @@ module.exports = {
     }
     return "any";
   },
+  hasDeclarationInFile(type, fileName) {
+    const symbol = type.getSymbol();
+    return Boolean(
+      symbol &&
+        symbol
+          .getDeclarations()
+          ?.some((declaration) => path.basename(declaration.getSourceFile().fileName) === fileName)
+    );
+  },
   isDocumentObject(node, context, fullTypeChecker) {
     if (fullTypeChecker) {
       const type = this.getNodeTypeAsString(fullTypeChecker, node, context);
diff --git a/lib/rules/no-inner-html.js b/lib/rules/no-inner-html.js
index 24b30f1..36e3fda 100644
--- a/lib/rules/no-inner-html.js
+++ b/lib/rules/no-inner-html.js
@@ -29,8 +29,19 @@ module.exports = {
     const fullTypeChecker = astUtils.getFullTypeChecker(context);
 
     function mightBeHTMLElement(node) {
-      const type = astUtils.getNodeTypeAsString(fullTypeChecker, node, context);
-      return type.match(/HTML.*Element/) || type === "any";
+      if (!fullTypeChecker) {
+        return true;
+      }
+
+      const tsNode = context.sourceCode.parserServices.esTreeNodeToTSNodeMap.get(node);
+      const tsType = fullTypeChecker.getTypeAtLocation(tsNode);
+      const type = fullTypeChecker.typeToString(tsType);
+      return (
+        type.match(/HTML.*Element/) ||
+        type === "any" ||
+        // A local type can also be named Element, so only trust the DOM library declaration.
+        (type === "Element" && astUtils.hasDeclarationInFile(tsType, "lib.dom.d.ts"))
+      );
     }
 
     return {
diff --git a/lib/rules/no-postmessage-star-origin.js b/lib/rules/no-postmessage-star-origin.js
index 5a76c0e..e1ab083 100644
--- a/lib/rules/no-postmessage-star-origin.js
+++ b/lib/rules/no-postmessage-star-origin.js
@@ -9,6 +9,15 @@
 
 const astUtils = require("../ast-utils");
 
+function isWindowOrAny(fullTypeChecker, type) {
+  const typeName = fullTypeChecker.typeToString(type);
+  return (
+    typeName === "any" ||
+    // A local type can also be named Window, so only trust the DOM library declaration.
+    (typeName === "Window" && astUtils.hasDeclarationInFile(type, "lib.dom.d.ts"))
+  );
+}
+
 module.exports = {
   meta: {
     type: "suggestion",
@@ -40,8 +49,12 @@ module.exports = {
             node.callee.object
           );
           const tsType = fullTypeChecker.getTypeAtLocation(tsNode);
-          const type = fullTypeChecker.typeToString(tsType);
-          if (type !== "any" && type !== "Window") {
+          // Unions such as Window | null must be checked one constituent at a time.
+          if (tsType.isUnionOrIntersection()) {
+            if (tsType.types.every((t) => !isWindowOrAny(fullTypeChecker, t))) {
+              return;
+            }
+          } else if (!isWindowOrAny(fullTypeChecker, tsType)) {
             return;
           }
         }
diff --git a/tests/fixtures/ts/tsconfig.json b/tests/fixtures/ts/tsconfig.json
index ed4dc2f..f33e424 100644
--- a/tests/fixtures/ts/tsconfig.json
+++ b/tests/fixtures/ts/tsconfig.json
@@ -1,3 +1,6 @@
 {
+  "compilerOptions": {
+    "strictNullChecks": true
+  },
   "include": ["estree.ts"]
 }
diff --git a/tests/lib/rules/no-inner-html.js b/tests/lib/rules/no-inner-html.js
index 34bde71..9bf3b72 100644
--- a/tests/lib/rules/no-inner-html.js
+++ b/tests/lib/rules/no-inner-html.js
@@ -30,6 +30,18 @@ ruleTester.run(ruleId, rule, {
         test.innerHTML = test;
         test.outerHTML = test;
       `
+    },
+    {
+      languageOptions: testUtils.tsLanguageOptions,
+      code: `
+        function main() {
+          class Element {
+            innerHTML = "";
+          }
+          const element = new Element();
+          element.innerHTML = "test";
+        }
+      `
     }
   ],
   invalid: [
@@ -48,6 +60,15 @@ ruleTester.run(ruleId, rule, {
         { messageId: "noInsertAdjacentHTML", line: 5 }
       ]
     },
+    {
+      languageOptions: testUtils.tsLanguageOptions,
+      code: `
+        function main(element: Element) {
+          element.innerHTML = "test";
+        }
+      `,
+      errors: [{ messageId: "noInnerHtml", line: 3 }]
+    },
     {
       code: `
         element.innerHTML = 'test';
diff --git a/tests/lib/rules/no-postmessage-star-origin.js b/tests/lib/rules/no-postmessage-star-origin.js
index 91df84f..a755c35 100644
--- a/tests/lib/rules/no-postmessage-star-origin.js
+++ b/tests/lib/rules/no-postmessage-star-origin.js
@@ -26,6 +26,19 @@ class WindowLike {
 function main() {
   var w: WindowLike = new WindowLike();
   w.postMessage('test', '*');
+}
+      `
+    },
+    {
+      languageOptions: testUtils.tsLanguageOptions,
+      code: `
+function main() {
+  class Window {
+    postMessage(): void {
+    };
+  }
+  const target = new Window();
+  target.postMessage('test', '*');
 }
       `
     }
@@ -52,6 +65,14 @@ function main() {
         { messageId: "default", line: 2 },
         { messageId: "default", line: 4 }
       ]
+    },
+    {
+      languageOptions: testUtils.tsLanguageOptions,
+      code: `
+        declare const target: Window | null;
+        target.postMessage(message, "*");
+      `,
+      errors: [{ messageId: "default", line: 3 }]
     }
   ]
 });