Alex has stated three things he'd like to do - strengthen the late-game bullpen options, look for a early-to-mid rotation starter, and try to find a middle of the order bat. I'll take these in reverse order for reasons that shall become apparent.
Middle of the order Bat:
A quick review of the roster construction makes it apparent that the only places to add a significant bat are, in order: first Base, DH, and LF
Ranking by OPS all hitters with at least 400 plate appearances who could reasonably be described as playing one of those positions, filtering out players we already have and players who are unavailable either because of prohibitive contracts or circumstances (such as being 5 & 10 players) or because of their value to the team (IMO) this is what I come up with (players with an OPS over .800 as befits a supposed "big bat"):
Mike Morse - his 2011 was an incredible outlier from the rest of his career. Although to be fair he showed some signs in 2010 in half as many plate appearances. Something is amiss because the Nationals GM called Adam LaRoche the presumptive 2012 first baseman, it may mean Morse sucks on defense. If true, one wonders if there's any place on the field they CAN hide him. He came essentially out of nowhere at 28, but that's a phenomena the Jays are not unacquainted with. Dude is 6'5" and 230, how it is he broke in as a SS I'll never know. If he can continue to hit like he has the last couple of years, he'd be conceivably a better DH than EE and a better (offensive) 1B than Lind if Lind doesn't rebound to his peak form. The Nats need outfield help.
Carlos Quintin - Had his career year, so far, at 25 and his three seasons since have been much less impressive. He has an .815 OPS over that span and frankly that's not much better than what one might reasonably expect of the in-house options. He'd have to be pretty cheap for me to be interested.
Seth Smith - pronounced home/road splits over his career.
Billy Butler - arguably best is yet to come, and has an .844 OPS over last three seasons. Only has 2 years of control left and will be trending towards costly to retain, but a legitimate target if the Royals think DH is easier to fill than the gaps in their rotation. Price uncertain.
Andrew McCutchen - Pirates have said they will listen, but will be wildly expensive i think. Might set up CF conflict.
I'm going to say McCutchen is too expensive and Quintin and smith are unimpressive. Either Morse or Butler would be intriguing, and Butler would better suit the time frame in terms of age if he can be reasonably extended. One might imagine the Royals would be interested in taking EE to stop-gap the DH position, or maybe they would see something of value in Cooper? Additionally they would surely be receiving a couple of interesting prospect and a filler or two. I'd try to start with Jimenez and see if they liked Jenkins, I wouldn't do the top shelf guys for a DH unless I thought he had David Ortiz ability.
Potential free agents that someone might see attractive (none for me thanks) are JD Drew, Magglio Ordonez, Johnny Damon, Luke Scott, Andruw Jones or Wilson Betimit.
(ranked by 2011 ERA)
James Shields - in division, won't happen.
Matt Cain - some speculate the Giants can't pay him and Lincicum too, but he's a free agent in one year which violate's Alex's "years of control"qualification. would be wildly expensive. Gose would have to be in the deal along with 2 or 3 other really good players, which I can't see doing for potentially just one season.
Matt Garza - everyone seems to recognize him as the most likely candidate, personally I'd liked the idea much better if we could have been trying to get him and Marshall in a package. There's more to like here but I'm not sure you sell the Cubs on lower level prospects which AA seems to be focused on dealing. Could you do it without giving up Gose or d'Arnaud? Because I'm thinking Alex wouldn't pay that price.
Brandon McCarthy - Oakland is clearing the decks, why not McCarthy too? Here again you are only getting one season, but the price should be much more affordable. Thing is, his 25 starts and 170 IP last year were career highs, which might mean he's made of glass but also might mean he hasn't had a chance to peak yet. Like Gio, his numbers were much more middle-of-the-pack away from that great home park and I'm not sure he's much if any better than the in-house options.
Felix Hernandez - I can't imagine he's actually available at any price.
Wandy Rodriguez - Would AA really commit to $36 million over the next two seasons? For age 33-35 years? My sources say no. if he wanted to do that he could just go on and sign Roy Oswalt to a deal just like it. and get a better value IMO, even before one considers the cost in prospects.
Zach Grienke - probably not available (why would they sign Ramirez if they are tearing it down) and would be very expensive in prospects for one year of service before free agency. An extension would be a massive contract.
IMO, the only POSSIBLE deal here, realistically, is Garza, and that only if Theo & Co. can be persuaded by a package of players below AA ball. Potential FA signings include Oswalt, Rich Harden, Edwin Jackason or Hiroki Kuroda.
Late Inning Relievers :
(grouped by team)
Jesse Crain and Matt Thornton - White Sox are shuffling the deck, Kenny Williams has already hooked up with AA twice on significant deals, Jays really need a late inning lefty (Thornton). Thornton is controlable for three years with the downside that the contract is a tad high and it's his age 35-37 seasons. Effectiveness slipped some in 2011 after three dominant years - start of a decline, or a blip? $12 mil is a lot for them to commit if they are not sure of the answer to that. Crain is signed for two more (age 30-31) at $4.5 each year which is certainly manageable. not sure the cost to get both, or the practicality of essentially buying the whole back-end of the White Sox 'pen, but if you got them both together it would theoretically solve the issue in one deal. (You could also look at bringing back Jason Frasor if better options failed)
Joakim Soria - (deserves separate listing) would set up a re-run of "this is our closer until we get another closer" routine from last year, but would theoretically set up a heck of an 8th/9th combo. Three years of control on the contract at manageable costs. Has turned down impressive offers before. Logically you wouldn't trade a position prospect like d'Arnaud or Gose for ANY reliever, but it might cost that. Rumor has it they like Rasmus but despite Colby's lost year, I'd be scared to death to trade a premium CF upside for a reliever, even a closer. And i doubt they would sweeten their end of the deal.
Greg Holland, Blake Wood, Tim Collins - Okay, Collins . . . I have no reason to think he's available, but given my emotional attachment I figure why not mention him? Holland has been mentioned as a potential target for the Jays, and Wood is another guy in a suddenly deep 'pen.
The royals had six good relievers last year and have added two free agent signings this off-season. someone will get pushed out. and while it's good to have depth, it also makes it possible the shop the guy or guys who're being underused. All these guys are pre-arb, cheap, and have years of control left.
Andrew Bailey - Very expensive, you'd be paying closer premium when you already have a closer (as with Soria above) - if you knew you were getting the guy from his rookie year . . . but are you? Still the very low walk rate is attractive. The very high asking price not so much.
Grant Balfour - No way Oakland hangs on to this guy, and he's much more affordable (in theory) prospect wise than Bailey. He'd fit right in to the back of the Jays 'pen.
Jeremy Affeldt - SF was rumored to be listening to offers for him in a cost cutting effort, would be the ideal 8th inning LH if you want a veteran back there. Can't pay too much for just one year though.
Sergio Romo - no reports that he's available, but would look very good in a Jays uniform.
Carlos Marmol - Cubs are rumored to be moving every valuable asset. Marmol has value but that walk rate worries me more than a little, even with all the strikeouts. it's trending in the right direction, but do you want to gamble that both he and Santos keep getting better in that department? Depends on the price in prospects I imagine.
There's enough fluidity in this market that I'd hate to make a prediction - but I'd be kinda surprised if one of these guys didn't end up on the Blue Jays roster by spring. Potential free agent alternatives include LH Mike Gonzalez, Darren Oliver and Damaso Marte, and RH David Ardsma, Taylor Buchholz (though not a proven late inning guy, his ratios were pretty good), Juan Cruz, and Dan Wheeler.