diff --git a/packages/eslint-plugin/docs/rules/promise-function-async.md b/packages/eslint-plugin/docs/rules/promise-function-async.md
index 4ca59ac341e..f3095ed1808 100644
--- a/packages/eslint-plugin/docs/rules/promise-function-async.md
+++ b/packages/eslint-plugin/docs/rules/promise-function-async.md
@@ -46,6 +46,7 @@ async function functionReturnsPromise() {
   return Promise.resolve('value');
 }
 
+// An explicit return type that is not Promise means this function cannot be made async, so it is ignored by the rule
 function functionReturnsUnionWithPromiseExplicitly(
   p: boolean,
 ): string | Promise<string> {
diff --git a/packages/eslint-plugin/src/rules/promise-function-async.ts b/packages/eslint-plugin/src/rules/promise-function-async.ts
index e8b9f44739b..34fbc7ee672 100644
--- a/packages/eslint-plugin/src/rules/promise-function-async.ts
+++ b/packages/eslint-plugin/src/rules/promise-function-async.ts
@@ -111,12 +111,12 @@ export default util.createRule<Options, MessageIds>({
       const returnType = checker.getReturnTypeOfSignature(signatures[0]);
 
       if (
-        !util.containsTypesByName(
+        !util.containsAllTypesByName(
           returnType,
           allowAny!,
           allAllowedPromiseNames,
-          // Only if a return type is explicitly declared must it match all
-          Boolean(node.returnType),
+          // If no return type is explicitly set, we check if any parts of the return type match a Promise (instead of requiring all to match).
+          node.returnType == null,
         )
       ) {
         // Return type is not a promise
diff --git a/packages/type-utils/src/containsTypesByName.ts b/packages/type-utils/src/containsAllTypesByName.ts
similarity index 69%
rename from packages/type-utils/src/containsTypesByName.ts
rename to packages/type-utils/src/containsAllTypesByName.ts
index 2e57984ac28..9c3fb522236 100644
--- a/packages/type-utils/src/containsTypesByName.ts
+++ b/packages/type-utils/src/containsAllTypesByName.ts
@@ -7,14 +7,14 @@ import { isTypeFlagSet } from './typeFlagUtils';
  * @param type Type being checked by name.
  * @param allowAny Whether to consider `any` and `unknown` to match.
  * @param allowedNames Symbol names checking on the type.
- * @param mustMatchAll Whether all parts have to match, as opposed to any parts matching.
+ * @param matchAnyInstead Whether to instead just check if any parts match, rather than all parts.
  * @returns Whether the type is, extends, or contains the allowed names (or all matches the allowed names, if mustMatchAll is true).
  */
-export function containsTypesByName(
+export function containsAllTypesByName(
   type: ts.Type,
   allowAny: boolean,
   allowedNames: Set<string>,
-  mustMatchAll: boolean,
+  matchAnyInstead = false,
 ): boolean {
   if (isTypeFlagSet(type, ts.TypeFlags.Any | ts.TypeFlags.Unknown)) {
     return !allowAny;
@@ -30,20 +30,20 @@ export function containsTypesByName(
   }
 
   const predicate = (t: ts.Type): boolean =>
-    containsTypesByName(t, allowAny, allowedNames, mustMatchAll);
+    containsAllTypesByName(t, allowAny, allowedNames, matchAnyInstead);
 
   if (isUnionOrIntersectionType(type)) {
-    return mustMatchAll
-      ? type.types.every(predicate)
-      : type.types.some(predicate);
+    return matchAnyInstead
+      ? type.types.some(predicate)
+      : type.types.every(predicate);
   }
 
   const bases = type.getBaseTypes();
 
   return (
     typeof bases !== 'undefined' &&
-    (mustMatchAll
-      ? bases.length > 0 && bases.every(predicate)
-      : bases.some(predicate))
+    (matchAnyInstead
+      ? bases.some(predicate)
+      : bases.length > 0 && bases.every(predicate))
   );
 }
diff --git a/packages/type-utils/src/index.ts b/packages/type-utils/src/index.ts
index a350d5c8658..dde032e1770 100644
--- a/packages/type-utils/src/index.ts
+++ b/packages/type-utils/src/index.ts
@@ -1,4 +1,4 @@
-export * from './containsTypesByName';
+export * from './containsAllTypesByName';
 export * from './getConstrainedTypeAtLocation';
 export * from './getContextualType';
 export * from './getDeclaration';