Can't hurt by trying. I can tell you that a bank may counter or they may be waiting for their asking price. You need to make the first move and see what the response is and then keep the dialogue going.
In most cases a bank will counter your offer at $1000 lass than asking and wait for your next move. Do not be insulted, just keep it going. You will arrive at a price that both sides will be happy with. If not move on.
As far as the Zestimate, Zillow uses sales from 6 mths ago and therefroe the price may not be a true indication of current market value.
If you are working with a Realtor, which I hope you are, (if not I can refer one in that area to you), have them prepare a CMA to give you the MOST current selling prices. Try to keep within 1 month. Base your offer and final price on that with room to increase if this is THE HOUSE. You may feel you are paying too much,however, unless you are going to sell within 2 years, you will not be. And also be aware that the cost to you is approx. $6. per 1000 mortgaged to get the home or not. Also make sure that if you walk away that you would not regret it later on.
And that's my story.
Good luck and may the blogs be with you.
First, toss out anything that zillow says, second, get ready for a 3-6 month process of playing the foreclosure game if not longer. Not only are you dealing with an overwhelmed banking system that is playing catch up to the massive amounts of foreclosures or short sales that they have on their books but also to a Real Estate system that is learning on how to deal with all this as well.
Be very careful with all these so called 'Foreclosure and short sale" experts. This is all still new to them as well.
If you have a lot of time on your hands then play the foreclosure game and you may get a good deal. If your looking to buy and move into a home in a shorter time frame like 1-3 months then find a nice home that is not bank owned (Yes, Their is a lot of them out there still) and make an offer after you have had a Realtor run the comps and done some homework for you.