diff --git a/packages/eslint-plugin/docs/rules/explicit-function-return-type.md b/packages/eslint-plugin/docs/rules/explicit-function-return-type.md
index fb8ad9a7ffc..e63d3d351a2 100644
--- a/packages/eslint-plugin/docs/rules/explicit-function-return-type.md
+++ b/packages/eslint-plugin/docs/rules/explicit-function-return-type.md
@@ -84,6 +84,20 @@ node.addEventListener('click', function() {});
 const foo = arr.map(i => i * i);
 ```
 
+### allowTypedFunctionExpressions
+
+Examples of additional **correct** code for this rule with `{ allowTypedFunctionExpressions: true }`:
+
+```ts
+type FuncType = () => string;
+
+let arrowFn: FuncType = () => 'test';
+
+let funcExpr: FuncType = function() {
+  return 'test';
+};
+```
+
 ## When Not To Use It
 
 If you don't wish to prevent calling code from using function return values in unexpected ways, then
diff --git a/packages/eslint-plugin/src/rules/explicit-function-return-type.ts b/packages/eslint-plugin/src/rules/explicit-function-return-type.ts
index e4f5dd46a3c..d767959c29f 100644
--- a/packages/eslint-plugin/src/rules/explicit-function-return-type.ts
+++ b/packages/eslint-plugin/src/rules/explicit-function-return-type.ts
@@ -4,6 +4,7 @@ import * as util from '../util';
 type Options = [
   {
     allowExpressions?: boolean;
+    allowTypedFunctionExpressions?: boolean;
   }
 ];
 type MessageIds = 'missingReturnType';
@@ -28,6 +29,9 @@ export default util.createRule<Options, MessageIds>({
           allowExpressions: {
             type: 'boolean',
           },
+          allowTypedFunctionExpressions: {
+            type: 'boolean',
+          },
         },
         additionalProperties: false,
       },
@@ -36,27 +40,41 @@ export default util.createRule<Options, MessageIds>({
   defaultOptions: [
     {
       allowExpressions: true,
+      allowTypedFunctionExpressions: false,
     },
   ],
   create(context, [options]) {
     /**
-     * Checks if the parent of a function expression is a constructor.
-     * @param parent The parent of a function expression node
+     * Checks if a node is a constructor.
+     * @param node The node to check
+     */
+    function isConstructor(node: TSESTree.Node): boolean {
+      return (
+        node.type === AST_NODE_TYPES.MethodDefinition &&
+        node.kind === 'constructor'
+      );
+    }
+
+    /**
+     * Checks if a node is a setter.
+     * @param parent The node to check
      */
-    function isConstructor(parent: TSESTree.Node): boolean {
+    function isSetter(node: TSESTree.Node): boolean {
       return (
-        parent.type === AST_NODE_TYPES.MethodDefinition &&
-        parent.kind === 'constructor'
+        node.type === AST_NODE_TYPES.MethodDefinition && node.kind === 'set'
       );
     }
 
     /**
-     * Checks if the parent of a function expression is a setter.
-     * @param parent The parent of a function expression node
+     * Checks if a node is a variable declarator with a type annotation.
+     * @param node The node to check
      */
-    function isSetter(parent: TSESTree.Node): boolean {
+    function isVariableDeclaratorWithTypeAnnotation(
+      node: TSESTree.Node,
+    ): boolean {
       return (
-        parent.type === AST_NODE_TYPES.MethodDefinition && parent.kind === 'set'
+        node.type === AST_NODE_TYPES.VariableDeclarator &&
+        !!node.id.typeAnnotation
       );
     }
 
@@ -103,6 +121,14 @@ export default util.createRule<Options, MessageIds>({
         return;
       }
 
+      if (
+        options.allowTypedFunctionExpressions &&
+        node.parent &&
+        isVariableDeclaratorWithTypeAnnotation(node.parent)
+      ) {
+        return;
+      }
+
       checkFunctionReturnType(node);
     }
 
diff --git a/packages/eslint-plugin/tests/rules/explicit-function-return-type.test.ts b/packages/eslint-plugin/tests/rules/explicit-function-return-type.test.ts
index 0abc6000f18..932e50e9714 100644
--- a/packages/eslint-plugin/tests/rules/explicit-function-return-type.test.ts
+++ b/packages/eslint-plugin/tests/rules/explicit-function-return-type.test.ts
@@ -92,6 +92,28 @@ function test() {
         },
       ],
     },
+    {
+      filename: 'test.ts',
+      code: `
+var arrowFn: Foo = () => 'test';
+            `,
+      options: [
+        {
+          allowTypedFunctionExpressions: true,
+        },
+      ],
+    },
+    {
+      filename: 'test.ts',
+      code: `
+var funcExpr: Foo = function() { return 'test'; };
+            `,
+      options: [
+        {
+          allowTypedFunctionExpressions: true,
+        },
+      ],
+    },
   ],
   invalid: [
     {
@@ -188,5 +210,29 @@ class Test {
         },
       ],
     },
+    {
+      filename: 'test.ts',
+      code: `var arrowFn = () => 'test';`,
+      options: [{ allowTypedFunctionExpressions: true }],
+      errors: [
+        {
+          messageId: 'missingReturnType',
+          line: 1,
+          column: 15,
+        },
+      ],
+    },
+    {
+      filename: 'test.ts',
+      code: `var funcExpr = function() { return 'test'; };`,
+      options: [{ allowTypedFunctionExpressions: true }],
+      errors: [
+        {
+          messageId: 'missingReturnType',
+          line: 1,
+          column: 16,
+        },
+      ],
+    },
   ],
 });