Yes, this is possible.
When you insert the HTML Editor instance on your page, Click on the
customize link in the Global File upload Settings section and enable file management. If you want to use advanced file management, you will need to have a database connection set up.
Make sure that the toolbar you select has the linking options available.
When viewing the page in a browser, click the link icon, you will; have the option to upload a file to the server when creating the link.