Seller can counter any way they like. If you made an offer, do you have a Realtor? If so, they need to be answering your questions. If you are going to make offers on REO (Real Estate Owned), foreclosures you should be educated on what to expect, like "As-Is" on the offer(s). Most REO's price them below market, to get multple offers (like what happened to you) and ensure a "quick" sale. If the offers are close, bank will take cash offer because there are less hassles to deal with on the loan and a quicker sale. However, like most sellers, in the end they want the strongest offer that nets them the most.
Best of luck.
Spirit Messingham, PLLC
3rd Generation Full-Time RealtorÂ®
Tierra Antigua Realty
Direct (520) 471-6900