JSR 335 spec, chapter "15.28.2 Run-time Evaluation of Method References" contains following assertion:

If the method reference has the form ExpressionName :: NonWildTypeArgumentsopt Identifier or Primary :: NonWildTypeArgumentsopt Identifier, the body of the invocation method has the effect of invoking the compile-time declaration of the method reference, as described in 15.12.4.3, 15.12.4.4, 15.12.4.5.[jsr335-15.28.2-41]

This assertions refers to chapter "15.12.4.3. Check Accessibility of Type and Method" which contains following assertions:

Let C be the class containing the method invocation, and let T be the qualifying type of the method invocation (��13.1), and let m be the name of the method as determined at compile-time (��15.12.3).
...
If m is private, then m is accessible if and only if C is T, or C encloses T, or T encloses C, or T and C are both enclosed by a third class.
...

According to these assertions MethodInvoker.invoke should succeed for following three cases: 

case 1:
class MethodInvoker {
    public static void invoke() {
        MethodSupplier ms = new MethodSupplier();
        MyFunctionalInterface1 fi = ms::m;
        fi.invokeMethodReference();
    }
    static class MethodSupplier {
        private void m() {
            System.out.println("m");
        }    
    }
}

case 2:
class MethodSupplier {
    private void m() {
        System.out.println("m");
    }    
    static class MethodInvoker {
        public static void invoke() {
            MethodSupplier ms = new MethodSupplier();
            MyFunctionalInterface2 fi = ms::m;
            fi.invokeMethodReference();
        }
    }
}

case 3:
class ThirdClass {
    static class MethodSupplier {
        private void m() {
            System.out.println("m");
        }    
    }

    static class MethodInvoker {
        public static void invoke() {
            MethodSupplier ms = new MethodSupplier();
            MyFunctionalInterface3 fi = ms::m;
            fi.invokeMethodReference();
        }
    }
}

but all of these 3 cases fail with following stack traces correspondingly:

case1:
Exception in thread "main" java.lang.reflect.InvocationTargetException

	at sun.reflect.NativeMethodAccessorImpl.invoke0(Native Method)
	at sun.reflect.NativeMethodAccessorImpl.invoke(NativeMethodAccessorImpl.java:57)
	at sun.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
	at java.lang.reflect.Method.invoke(Method.java:491)
	at Test1.main(Test1.java:20)
Caused by: java.lang.IncompatibleClassChangeError
	at java.lang.invoke.MethodHandleNatives.linkMethodHandleConstant(MethodHandleNatives.java:383)
	at MethodInvoker.invoke(Test1.java:8)
	... 5 more
Caused by: java.lang.IllegalAccessException: member is private: MethodInvoker$MethodSupplier.m()void/invokeSpecial, from MethodInvoker
	at java.lang.invoke.MemberName.makeAccessException(MemberName.java:744)
	at java.lang.invoke.MethodHandles$Lookup.checkAccess(MethodHandles.java:1129)
	at java.lang.invoke.MethodHandles$Lookup.checkMethod(MethodHandles.java:1092)
	at java.lang.invoke.MethodHandles$Lookup.getDirectMethodCommon(MethodHandles.java:1203)
	at java.lang.invoke.MethodHandles$Lookup.getDirectMethod(MethodHandles.java:1193)
	at java.lang.invoke.MethodHandles$Lookup.linkMethodHandleConstant(MethodHandles.java:1285)
	at java.lang.invoke.MethodHandleNatives.linkMethodHandleConstant(MethodHandleNatives.java:381)
	... 6 more

case 2:
Exception in thread "main" java.lang.reflect.InvocationTargetException
	at sun.reflect.NativeMethodAccessorImpl.invoke0(Native Method)
	at sun.reflect.NativeMethodAccessorImpl.invoke(NativeMethodAccessorImpl.java:57)
	at sun.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
	at java.lang.reflect.Method.invoke(Method.java:491)
	at Test2.main(Test2.java:21)
Caused by: java.lang.IncompatibleClassChangeError
	at java.lang.invoke.MethodHandleNatives.linkMethodHandleConstant(MethodHandleNatives.java:383)
	at MethodSupplier$MethodInvoker.invoke(Test2.java:13)
	... 5 more
Caused by: java.lang.IllegalAccessException: member is private: MethodSupplier.m()void/invokeSpecial, from MethodSupplier$MethodInvoker
	at java.lang.invoke.MemberName.makeAccessException(MemberName.java:744)
	at java.lang.invoke.MethodHandles$Lookup.checkAccess(MethodHandles.java:1129)
	at java.lang.invoke.MethodHandles$Lookup.checkMethod(MethodHandles.java:1092)
	at java.lang.invoke.MethodHandles$Lookup.getDirectMethodCommon(MethodHandles.java:1203)
	at java.lang.invoke.MethodHandles$Lookup.getDirectMethod(MethodHandles.java:1193)
	at java.lang.invoke.MethodHandles$Lookup.linkMethodHandleConstant(MethodHandles.java:1285)
	at java.lang.invoke.MethodHandleNatives.linkMethodHandleConstant(MethodHandleNatives.java:381)
	... 6 more

case 3:
Exception in thread "main" java.lang.reflect.InvocationTargetException
	at sun.reflect.NativeMethodAccessorImpl.invoke0(Native Method)
	at sun.reflect.NativeMethodAccessorImpl.invoke(NativeMethodAccessorImpl.java:57)
	at sun.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
	at java.lang.reflect.Method.invoke(Method.java:491)
	at Test3.main(Test3.java:24)
Caused by: java.lang.IncompatibleClassChangeError
	at java.lang.invoke.MethodHandleNatives.linkMethodHandleConstant(MethodHandleNatives.java:383)
	at ThirdClass$MethodInvoker.invoke(Test3.java:16)
	... 5 more
Caused by: java.lang.IllegalAccessException: member is private: ThirdClass$MethodSupplier.m()void/invokeSpecial, from ThirdClass$MethodInvoker
	at java.lang.invoke.MemberName.makeAccessException(MemberName.java:744)
	at java.lang.invoke.MethodHandles$Lookup.checkAccess(MethodHandles.java:1129)
	at java.lang.invoke.MethodHandles$Lookup.checkMethod(MethodHandles.java:1092)
	at java.lang.invoke.MethodHandles$Lookup.getDirectMethodCommon(MethodHandles.java:1203)
	at java.lang.invoke.MethodHandles$Lookup.getDirectMethod(MethodHandles.java:1193)
	at java.lang.invoke.MethodHandles$Lookup.linkMethodHandleConstant(MethodHandles.java:1285)
	at java.lang.invoke.MethodHandleNatives.linkMethodHandleConstant(MethodHandleNatives.java:381)
	... 6 more

This was reproduced on b100, platform: Windows7x64.

The minimized tests are attached in a single archive.

In order to reproduce case 1, 2 and 3 please run Test1.main, Test2.main and Test3.main from attached archive correspondingly.