In May, he said he would want to "deal with these issues on a ... permanent basis as opposed to a stopgap effort that would require unraveling and re-evaluation," he said.

Romney opposes the nearly $500 billion in cuts to defense spending called for under the so-called sequester. And he doesn't want the Bush-era tax cuts to expire for anyone.

At the same time, he has promised to balance the budget in eight to 10 years without cutting defense or raising taxes.

Best-case scenario

Lawmakers and policy experts say the best a lame-duck Congress can probably do before the end of the year is come up with a "bridge" or "framework" deal.

Broadly, that might mean lawmakers postpone fiscal cliff measures for several months to a year, negotiate a small amount of spending cuts and agree to strike a large deficit-reduction deal by a date certain next year, perhaps according to agreed-upon targets for spending and revenue.

But then if lawmakers failed to deliver next year, the fiscal cliff or something similar would take effect.

- Jessica Yellin, CNN's senior White House correspondent, contributed to this report.