diff --git a/source/readonly-deep.d.ts b/source/readonly-deep.d.ts
index f4ad9e3ff..9fa397caa 100644
--- a/source/readonly-deep.d.ts
+++ b/source/readonly-deep.d.ts
@@ -34,8 +34,14 @@ data.foo.push('bar');
 @category Set
 @category Map
 */
-export type ReadonlyDeep<T> = T extends BuiltIns | ((...arguments: any[]) => unknown)
+export type ReadonlyDeep<T> = T extends BuiltIns
 	? T
+	: T extends (...arguments: any[]) => unknown
+	? {} extends ReadonlyObjectDeep<T>
+		? T
+		: HasMultipleCallSignatures<T> extends true
+		? T
+		: ((...arguments: Parameters<T>) => ReturnType<T>) & ReadonlyObjectDeep<T>
 	: T extends Readonly<ReadonlyMap<infer KeyType, infer ValueType>>
 	? ReadonlyMapDeep<KeyType, ValueType>
 	: T extends Readonly<ReadonlySet<infer ItemType>>
@@ -62,3 +68,18 @@ Same as `ReadonlyDeep`, but accepts only `object`s as inputs. Internal helper fo
 type ReadonlyObjectDeep<ObjectType extends object> = {
 	readonly [KeyType in keyof ObjectType]: ReadonlyDeep<ObjectType[KeyType]>
 };
+
+/**
+Test if the given function has multiple call signatures.
+
+Needed to handle the case of a single call signature with properties.
+
+Multiple call signatures cannot currently be supported due to a TypeScript limitation.
+@see https://github.com/microsoft/TypeScript/issues/29732
+*/
+type HasMultipleCallSignatures<T extends (...arguments: any[]) => unknown> =
+	T extends {(...arguments: infer A): unknown; (...arguments: any[]): unknown}
+		? unknown[] extends A
+			? false
+			: true
+		: false;
diff --git a/test-d/readonly-deep.ts b/test-d/readonly-deep.ts
index 5dcb2850c..d2c45ae9d 100644
--- a/test-d/readonly-deep.ts
+++ b/test-d/readonly-deep.ts
@@ -1,11 +1,29 @@
 import {expectType, expectError} from 'tsd';
 import {ReadonlyDeep} from '../index';
+import {ReadonlyObjectDeep} from '../source/readonly-deep';
+
+type Overloaded = {
+	(foo: number): string;
+	(foo: string, bar: number): number;
+};
+
+type Namespace = {
+	(foo: number): string;
+	baz: boolean[];
+};
+
+type NamespaceWithOverload = Overloaded & {
+	baz: boolean[];
+};
 
 const data = {
 	object: {
 		foo: 'bar',
 	},
 	fn: (_: string) => true,
+	fnWithOverload: ((_: number) => 'foo') as Overloaded,
+	namespace: {} as unknown as Namespace,
+	namespaceWithOverload: {} as unknown as NamespaceWithOverload,
 	string: 'foo',
 	number: 1,
 	boolean: false,
@@ -28,6 +46,9 @@ const readonlyData: ReadonlyDeep<typeof data> = data;
 
 readonlyData.fn('foo');
 
+readonlyData.fnWithOverload(1);
+readonlyData.fnWithOverload('', 1);
+
 expectError(readonlyData.string = 'bar');
 expectType<{readonly foo: string}>(readonlyData.object);
 expectType<string>(readonlyData.string);
@@ -46,3 +67,14 @@ expectType<Readonly<ReadonlyMap<string, string>>>(readonlyData.readonlyMap);
 expectType<Readonly<ReadonlySet<string>>>(readonlyData.readonlySet);
 expectType<readonly string[]>(readonlyData.readonlyArray);
 expectType<readonly ['foo']>(readonlyData.readonlyTuple);
+
+expectType<((foo: number) => string) & ReadonlyObjectDeep<Namespace>>(readonlyData.namespace);
+expectType<string>(readonlyData.namespace(1));
+expectType<readonly boolean[]>(readonlyData.namespace.baz);
+
+// These currently aren't readonly due to TypeScript limitations.
+// @see https://github.com/microsoft/TypeScript/issues/29732
+expectType<NamespaceWithOverload>(readonlyData.namespaceWithOverload);
+expectType<string>(readonlyData.namespaceWithOverload(1));
+expectType<number>(readonlyData.namespaceWithOverload('foo', 1));
+expectType<boolean[]>(readonlyData.namespaceWithOverload.baz);