You can offer whatever you want, and the agent has to do write it up, but you want to be careful not to insult the seller. How would you like it, if you sweated and toiled over your home for years and someone walks in and offers you 50% off of what it is worth.
The Indiana FHA Expert
I often do sort by square feet after getting a list, and look for a home with much larger square footage for the money in the same area.
Recent sold prices give you the best indication of where your offer should be for a home. Other things play into it (condition of home, major upgrades like kitchen/bath). If the seller is an estate sale that can really work for you. If the seller is distressed that helps too.
Listing price is determined by gathering hard data of comparable properties in that same community within the last 90 days (in today's softer market). When you look at a property for sale, ask your own buyer's agent to tell you the comps for that community (price per sq. ft.), then multiply by the sq. ft. Not all sellers price the home to sell, so you need access to your own hard data as a buyer.