There are 43 players on the roster (after eliminating the four FA-to-be). They probably want to clear about 10-11 of these guys, one way or another, to make room for seven or eight prospects.
Possible deletions:
Outrights: Appel, Therrien, Kelly, Valentin, Florimon, Perkins.
Non-Tenders: Alvarez, Siegrist.
Trades: Rupp, Galvis, Joseph.
Possible prospects to protect:
Pitchers: Franklyn Kilome, Jose Taveras, Seranthony Dominguez, Miguel Nunez
Catcher: Deivi Grullon
Infielders: Mitch Walding
Outfielders: Carlos Tocci, Jiandido Tromp, Andrew Pullin.
I'm not convinced about any of the non-pitchers listed, though. None of them is an obvious breakout candidate (Walding, maybe?), and any of them would be hard to carry on a ML bench.
Prospects on nycphils' list I've left off: Saurez, Viza, Leibrandt. Saurez is in high-A ball, didn't dominate. Viza and Leibrandt just don't look good enough to worry about. McKenzie Mills isn't eligible this year - 2014 HS signee, signed at age 18.
Overall, this looks like it's eminently manageable - several of the prospects I would have worried about losing are already on the 4o-man roster, thanks to in-season turnover and promotions.