WASHINGTON — House Speaker John Boehner declared today that President Barack Obama was “damaging the presidency” with his unilateral action on immigration. He said the Republican-run House will not stand by, but gave no hint of what the response would be.

“I will say to you, the House will, in fact, act,” Boehner, R-Ohio, said at a news conference the morning after Obama announced plans to offer deportation relief and work permits to 5 million immigrants who are in the country illegally.

“We will listen to the American people, we will work with our members and we will work to protect the Constitution of the United States,” Boehner said.

But Republicans have few good options as they scramble for a solution that satisfies irate conservatives without alienating moderates, Hispanics and other voters who will be crucial for the 2016 presidential election. Possibilities include suing Obama or trying to fight his moves through the budget process.

The situation poses a major challenge for Boehner and incoming Senate Majority Leader Mitch McConnell, R-Ky., barely two weeks after midterm election victories that handed Republicans control of the Senate and increased the party’s majority in the House.

Obama’s move forces them to inaugurate their newly minted congressional majorities amid frantic GOP infighting that party leaders wanted to avoid.

With Republicans seething over Obama’s go-it-alone approach on such a contentious issue, it’s an open question whether Boehner and McConnell will be able to rein in the tea party faction in Congress that forced a politically damaging government shutdown a year ago over the president’s health care law.