I think most people who care have already figured these numbers out, but it's nice to have it down in print. I am referencing an old post of mine for some calculations. It's worth reading the last section about Joe Johnson at the very least.
What's the maximum contract for James, Bosh, or Johnson?
A max contract for James or Bosh depends on the salary cap. The largest contract they can receive from the bulls is either a 5% raise from their last contract or 30% of the total salary cap. If the salary cap is greater than $55,229,693, than they'll receive 30% of the salary cap; otherwise, if it's less, they'll receive a 5% raise. A 5% raise from their current contract would be $16,568,908, which represents the minimum amount for a maximum contract.
Joe Johnson's salary is much less than James or Bosh right now, so a 5% raise will almost certainly be less than 30% of the salary cap. For those curious Boozer's maximum deal would also be 30% of the salary cap.
What's the state of our cap?
Assuming we renounce everyone, we have 5 players under contract. Rose, Deng, Johnson, Gibson, Noah ($22,850,976). We also have 6 cap holds that are applied to make sure we fill out our roster ($473,604 *6). Which means our committed salary stands at $25,692,600.
If we signed another FA, then we would only count 5 cap holds and our committed salary would be $25,218,996 + FA contract.
Can we sign 2 FAs to maximum deals?
The short answer is no. A quick calculation:
Obviously, the only way to do this is if the salary cap is large. If the salary cap is over $55.3 million, then we would be signing two players to contracts worth 30% of the cap each. The value of all of our contracts would have to represent 40% of the salary cap, and the cap would have to be $63,047,490 ($25,218,996 / 0.40). Sadly, that isn't going to happen.
Does this Joe Johnson rumor have any legs to stand on?
K_yle33 mentions a rumor here that the Bulls may have agreed to sign Johnson to a contract for $60 million over 4 years (Thonus mentions it as well). This could be a great move if it made the bulls even more attractive to either LeBron or Bosh, but it obviously only works if we can actually sign them. A $60 million dollar contract over 4 years would start at $13,392,857.
If we take our $25,218,996 (committed salary) + $13,392,857 (contract for JJ)+ $16,568,908 (the minimum max starting salary) = $55,180,761. This figure is about $1 million below the often quoted $56.1 million cap, which is a good buffer in case that figure is high. Because these numbers work out so well, I think this rumor might be true, not to mention that we have no shooting guard to speak of. The sad news is that there probably wouldn't be enough money for Asik to come over this year, unless he's willing to take a pay cut. That being said, the Bulls would probably have the best starting 5 of any team with Rose, Johnson, Deng, Bosh/Bron, Noah (with Gibson and James Johnson on the bench).