I believe signing bonuses are set for certain slots, and most teams offer slot money.
You can be offered more than slot money, which happens a lot, but I believe you have to get league approval to do so, though they usually approve. I am not sure if that's exactly how it works, but I know it's something like that.
For instance, our signee, David Renfroe, got drafted in the third round. Third round money is definitely less than a million, but the Red Sox are rumored to be considering an offer of up to $2M or $3M, definitely over $1M at least. I know Scott Van Slyke (Andy's son), a signee of ours a few years back was asking for a specific round to be drafted, or he was coming to school. He got drafted well after that particular round, but the team that drafted him offered him the money that he'd have gotten for the round he wanted, and he signed with the team. In that case, I believe that particular team had missed on signing a few of its top picks and offered him extra to get him on.
I also know that often players will put out publicly what it will take to sign them, and often they will drop well down the board because teams know they are basically unsignable without offering more than slot money if they don't want them at a particular round. For an example, Paul Maholm came out of high school saying he was going to MSU if he didn't get a $1M signing bonus. He was projected I believe as a 4th rounder on talent, but he dropped in the draft to the mid-teens of rounds because teams knew that if they signed him in the 4th or 5th round, they'd have to go above slot money to get him to sign, and they didn't think he was worth the risk of the $1M investment right off.
Obviously, that worked out well for Paul, as I believe he was a Top 10 pick coming out of State and got something like $2.5M for his signing bonus, and now he's one of the top starters for the Pirates.