Welcome to Android Central! Check the settings of the downloader app to see if there's an option to select the default save location. Make sure it's set to Internal Storage.
Incidentally, is this one of those "free" MP3 downloader apps? If so, be very careful. These apps skirt copyright laws by scouring the web for filesharing servers that have these music files (which may themselves have been uploaded illegally). The terms and conditions I've seen for some of these downloader apps specifically state that they're for "research" only, and that once you've finished "researching" a song, you must delete it from your phone. So in other words, if you keep the song (assuming it isn't public domain), it's illegal. Apart from that, you're also exposing yourself to a big risk of malware, since those servers are unlikely to be vetted for malware.