Biagini is under club control for awhile. Would the Jays want to part with him (especially if guys like Giles, Hudson and Phelps will be on the move, I mean, you have to have somebody to play the games)?
One thing I'm really curious about is how far management goes with trading off current-year assets. Will they even dip into some of the guys they have some control over?
Because chances are that he doesn't get better than he is right now and if he has value in the form of prospects, you do it. Especially when bullpen guys are so volatile and he could easily fall off of a cliff next year and become worthless. If the Braves are willing to give up a decent prospect or two, or package him with Giles for one of their 2nd tier pitchers and maybe something extra, the Jays will not think twice. They have been accumulating a bunch of high minors pitchers through trades, and they have guys worth a look to see what they have. Maybe they even go out and acquire some reclamation pieces or salary dumps for more assets to fill some holes. Those guys are more likely to fit the Jays' window/needs long term than Biagini.
As many others with a lot more knowledge than me have said, anything that is not a young prospect with upside is available... Which essentially means only guys like Vlad Jr., Bichette, Jansen and Biggio are off limits. Of course there are a long list of others who likely won't get enough on the open market to warrant trading (Gurriel Jr., Hernandez, McKinney, Grichuk, Drury, Tellez, all of the mid-20's pitchers outside of Biagini, etc.), but if some team randomly decided to give the Jays a good offer for, say, Hernandez, he's gone.