Couple of things first:
1. I think that Amare is the most readily and immediately available of the big men right now.
2. He also happens to be a great fit for what the Bulls do. He's a great PnR player and can hit the open 15 foot jumper as well as any PF in the league. He's also one of the better PF finishers at the rim in the league. The Bulls biggest weaknesses at the PF position on offense are Tyrus' inability to finish at the rim and his inability to really consistently hit his jumpers.
3. If the Bulls don't really like how Amare fits or are scared about his injury history, they can keep him and hope he opts out so that they have 2010 cap space. Alternatively, if Amare doesn't opt out, they could try to get the Raptors to bite on a sign-and-trade with Amare as the main piece for Bosh. If none of that happens, they have Amare's cap space on the books for 2010 and it takes them out of the running for the big FAs, but they might have space for the 2011 off-season.
4. A bird in hand is worth two in the bush.
The problems with Amare are well documented (lack of rebounding, defense, eye and knee injuries, exaggerated ego, etc.) so I won't get into them beyond that paranthetical notation. Here's the deal I would suggest from the Bulls to the Suns:
Suns send: Amar'e Stoudemire and Taylor Griffin
I have a great lack of faith in the Bulls' ability to actually sign anyone this off-season or to sign the right person, which is why I'd like this deal. Additionally, nothing about this deal would prevent them from selling off Hinrich and Salmons for expirings to re-open cap space for next summer, so that they could make a run at one of the other FAs.
I value certainty pretty highly and having Amare in hand (for all his flaws) would be pretty awesome and would make this year more fun to watch. It would also provide the Bulls with a pretty fierce starting 5 of Rose, Hinrich, Deng, Amare and Noah. Taj, Salmons, Devin and Taylor would be a pretty thin bench, but it could work for this season and it's not really that much thinner that what the Bulls have currently been working with. So, Gar, if you're listening, go get Amare (but only if you can do it on the cheap, like this).