Are you working with a real estate professional? This is something he/she should be able to assist you with. The beast offer below the "asking price" is one that can be justified and defended. With this said, it would be important to have a clear appreciation for the recently sold similar property in this location.
Banks base their asking prices on appraisals and BPO's(broker price opinions) but this doesn't necessarily mean they can't be wrong.
It sounds like you have made an offer that was not accepted and you are now as a point of considering making a second offer.......if this is the case, we would recommend making an offer that you can back up with solid information. If the home requires a new heating system or the wireing is faulty, make sure you submit a professional opinion and estimate for the work. If you have examples of similar homes that have sold for lower amounts, be sure to include this information as well. Your agent should be able to help you put together a package that may be helpful to your cause.
Unfortunately, you don't know what it's worth. It could be worth $75,000. It could be worth $125,000. You don't know. But you need to in order to counter the bank's offer.
Sounds as if the bank won't be too flexible. It's listed for $114,000. You offered $90,000. The bank countered at $113,000. The bank might not go any lower. But, really, that doesn't matter. What matters is what it's worth. If, hypothetically, your Realtor (you do have your own Realtor, I hope) says the house is worth $120,000, then take the bank's offer of $113,000. If your Realtor says it's only worth $105,000, then counter at some figure below $105,000.
But, as you can see, the one necessary number is: What's the house worth?
Ask your Realtor to run a CMA on the property. Then pay no more than that figure.
Hope that helps.