Can a house be sold as a "short sale" to a relative?
A short sale requires the active permission of the lender, because the buyer will not accept title (in other words, won't pay the purchase price) if the title still contains the lender's mortgage lien. If the lender does not approve the sale, it will require that the entire amount owed on the loan be paid before it will remove its mortgage from the title.
A short sale involves the lender's agreement to accept less than it is owed. The documentation is very detailed, and the lender retains the right to reject a proposed sale if the lender does not believe that it is receiving the current value of the property (even if that value is less than what is owed).
Whether or not the proposed buyer is a relative, the lender will very carefully examine the terms of the sale, as well as the original borrower's (seller now) payment history and current financial circumstances, before agreeing to accept an amount less than it is owed in exchange for releasing its mortgage. It might be particularly hazardous where the borrower is attempting to sell to a relative, and may be giving that relative a break on the price.
Keep in mind also that it is a federal felony to defraud a lender, which in this case might include the borrower being perceived as trying to avoid paying a legitimate debt by arranging a friendly transaction with a relative. This issue would be particularly important if the relative were a close family member, a child or spouse rather than a third cousin.