I don't believe that's allowed, to be honest, and so I don't think Miami would do it. What Barkley would do is the same thing all the other underclass does, send in their tape to be reviewed and graded by the Advisory Committee. They'll review it and let him know if they see him as a high 1st rounder or whatever.
From what I've heard, Matt Barkley wants the choice to be 'easy'. I think there are some smart people out there that like Barkley for the #2 QB ranking behind Luck, but not enough of them. Right now there are still a lot of Landry Jones fans and recently I heard that the Broncos are falling in love with Brandon Weeden as much as I do (I have Weeden #2 behind Luck on pure talent).
Barkley isn't the kind of guy that is going to find things being 'easy' for him in the Draft, not unlike Drew Brees back in 2001. Not unlike a T.J. Yates a year ago. How he impresses you is with his depth and his detail, but most NFL people want to be wowed by physical characteristics and bullet throws.