The law could also have broader economic consequences. Business leaders, for example, have said they worry about attracting workers if New York and other cities become even more expensive than lower-tax areas.

State leaders are still figuring out their response to the new law, and few have yet endorsed specific proposals. But they are moving quickly. Gov. Andrew M. Cuomo of New York, a Democrat, recently said he expected to provide a more detailed plan when he presented his state budget in mid-January.

“They want to target us for certain provisions?” Mr. Cuomo asked at a recent news conference. “Well, let’s see if we can redesign our tax code to get out of the federal trap that they set.”

Mr. Cuomo fired one of the first shots when he signed an executive order that let New Yorkers prepay their 2018 property taxes in 2017, before the new deduction cap takes effect. Several other state and local governments followed suit.

But in an indication of the hard road ahead for Democrats, the Internal Revenue Service issued guidance on Wednesday limiting the prepayment option. And the option was only a temporary reprieve — at best, homeowners could delay the impact by a single year.

State leaders are looking for longer-term solutions. Some have raised the possibility of shifting away from taxes on individuals toward taxes on corporations, which are still fully deductible under federal law. But that could cause its own problems: Raising taxes on businesses could make it harder for those states to compete for companies and jobs.

Other lawmakers have floated the idea of seeking out new sources of revenue, perhaps by legalizing — and taxing — marijuana.