There could be several reasons the seller would go back to their bank prior to accepting the offer.
The Seller may owe the bank more than they might net from the sale. The Seller may be asking the bank to participate in the loss with them. This situation is more likely to happen if the Seller is already late with their payments, otherwise their is less incentive for the bank to accept such terms.
Recently, a client of mine decided to sell their home after just 3 years of ownership. The house may be worth a little more than they paid for it, however the Loan they secured at purchase includes an early termination fee of 5% (usually a percentage of the loan amount) that would be due to the bank at Settlement. This Seller was advised to open dialogue with the bank immediately. The bank agreed to reduce the fee so the Seller would not have to pay more than what was received. The Seller will have to contact the Bank when an offer comes in.
Sometimes the lender is not a bank or mortgage company but a relative, friend or investor who shares in the profit if any. The Seller would be looking for their blessing.
I hope you get the idea from these examples.