Put in your best offer and wait till the appraisal comes back.
You can always bid No Appraisal contingency, but have a loan contingency.
The delta between the Appraisal Value and the offer price can be Brought in
by you to make up the difference.
I say offer what you are comfortable with and when the appraisal comes back, you can try to negotiate at that time. If you try to negotiate now, they will pick the highest offer of course. Good luck!
If the house doesn't appraise you can always ask the priced be reduced or pay the difference. It's a tough call but with inventory low if the house is right, 10k difference isn't much to pay over what the appraisal is to find a home you are happy with. I say go for it!
The Carrabba Group
Keller Williams Hollywood Hills
If it's of any value to you, I overpay for just about everything, and I usually get what I want. When I tell people what I paid for my first house in 1988, the story has no dramatic value, because it doesn't matter - it's worth four times that today, even after the crash.
Point being - this home is going to be the stage setting for the next several years of your life. Would you rather give it a try, or just let it go without a struggle?