Don't negotiate payment when coming up with a lease deal. You need to know the selling price (you want this as low as possible), the residual value or residual percentage (you want this as high as possible), the interest rate (for a lease, this is often known as the money factor; want this as low as possible), down payment, term, and allowable mileage. Negotiate the selling price, interest rate, and residual value. From these factors, you can calculate the lease payment (google "lease calculator" if you want an easy calculator to do this for you).
I agree with previous posters that putting money down on a lease is a bad idea. The only reason I might consider it on a FFE is if there was a gov't incentive or rebate that I would be using to cover the down payment.
GONE - 2015 FFE, Magnetic with leather interior, debadged & tinted
38,500 miles (since May 2015), average 246 Wh/mile
REPLACED WITH - 2018 Tesla Model3, LR RWD, Obsidian Black
20,000 miles (since Nov 2018), average 235 Wh/mile