"The best possible outcome is to take something like Simpson-Bowles as it came out originally and work off that," he said, of a deal to avoid the automatic tax hikes and spending cuts that go into effect at the end of the year.
But he said that reaching a final agreement won't be an easy process, since the president believes he has a mandate following the election while House Republicans believe they, too, have a mandate.
"I'm not at altogether clear how much control (Speaker) Boehner has over the overall caucus," Greenspan said. "At the end of the day it will all work out but it's going to be a bit hairy before we get there."
(Read More: Forget the 'Fiscal Cliff,' Look at These Cliffs.)
Greenspan also warned there could be significant implications for markets if a deal is not reached.