Most car dealers in Illinois have "yo yo" provisions in their contract, making their obligation to sell conditioned on a finance company agreeing to purchase the contract. They do this so they can give immediate possession of the car to the consumer, in the hope that they will go along with any changes in the deal. Illinois courts have upheld such deals, although they are inherently overreaching and rife with deception. The problem is that the dealer keeps part of the "finance charge" you agree to pay, so that the dealer's alleged inability to sell the contract often means that the finance company thinks the dealer wants to put too much in its pocket. Sometimes the dealer just tries to hold the consumer, who has taken a liking to the car, up for more money. Often this is compounded by a claim that the tradein has already been disposed of.
Legally, if the dealer cannot sell the contract, it can either (a) ask that you return the car and get back any downpayment/ tradein, without deduction of any sort, or (b) waive the provision and accept your payments yourself. The one thing that the dealer cannot force you to do is sign a new contract with more onerous terms.
I assume that what you meant by "change the amount financed" means that the finance company wants to reduce the amount it is willing to pay the dealer. "Amount financed" also has a technical meaning; it is one of the disclosure boxes on a Truth in Lending statement.
If the dealer is willing to give you a credit for the difference between what the finance company was asked to pay and what it was willing to pay, thereby reducing the price and payments, without a higher annual percentage rate, the contract probably allows the dealer to do this. But if any significant term of the contract is changed to your disadvantage, you cannot be compelled to agree to the new terms.
At this point, you should consult a consumer lawyer familiar with this sort of issue. They will want a clear copy of the frong and back of all papers you signed or received from the dealer. You should consult an attorney before agreeing to anything with the dealer.