I do not think that schools “expect” you to negotiate. And from what I understand, they cringe at the idea of anyone calling it a “negotiation.”
If you need to appeal to explain why you really need more money to make it work for your child to attend, then there is absolutely a process at most schools.
However, if the hope is just to “get a good deal” (or “negotiate”), then I think that will not be super well-received. You’re not buying a used car. :). Remember that behind these FA dollars are families who are generously donating their hard-earned money to the school. Their generosity should be honored, in my opinion.