Do you have the WP Security Scan plugin enabled? If so, disable it and see if it works. You can always re-enable WP Security Scan (a fabulous plugin) when you need it again later. I enable it every few weeks to make sure my settings are still secure on client sites.
I had a similar problem and it turned out that the theme I was using had some dodgy code in it’s functions.php. Try switching to the default theme and trying again, see if that makes a difference.