The U.S. Postal Service is projected to lose $8.3 billion this year, and, "On Sept. 30, I won't be able to pay my bills," Donahoe told USA Today.
On that date, a $5.5 billion payment is due to cover future retirees' health benefits.
Congress mandated six-day delivery in 1983, but Sens. Tom Carper, D-Del., and Susan Collins, R-Maine, have introduced legislation to allow the end of Saturday mail.
Donahoe estimates that step would save $3.1 billion annually and said it has "a much better chance today than a year ago. I don't know if I'd say 'likely' yet."
Looking ahead, he said, "At some point, we'll have to move to three" days a week, maybe in 15 years.
Sen. Jon Tester, D-Mont., says Saturday delivery is vital to rural areas.
"It's important to have delivery on Saturday in places like Montana to get things like medicine and things they need," he said.