PHOENIX — Bank of America and GMAC are firing up their formidable foreclosure machines again today, after a brief pause.

But hard-pressed homeowners like Lydia Sweetland are asking why lenders often balk at a less disruptive solution: short sales, which allow owners to sell deeply devalued homes for less than what remains on their mortgage.

Ms. Sweetland, 47, tried such a sale this summer out of desperation. She had lost her high-paying job and drained her once-flush retirement savings, and her bank, GMAC, wouldn’t modify her mortgage. After seven months of being unable to pay her mortgage, she decided that a short sale would give her more time to move out of her Phoenix home and damage her credit rating less than a foreclosure.

She owes $206,000 and found a buyer who would pay $200,000. Last Friday, GMAC rejected that offer and said it would foreclose in seven days, even though, according to Ms. Sweetland’s broker, the bank estimates it will make $19,000 less on a foreclosure than on a short sale.