They were also coming to another realization about the pitfalls of their power, and it was the same reckoning that Democrats had when they decided to block the Homeland Security bill: Republicans, as the party that controls Congress, would be blamed as the department closed in on the Feb. 27 deadline when it will run out of money.

“I’m hoping public opinion starts to recognize that it’s not Republicans who are objecting or obstructing,” said Senator Ron Johnson, Republican of Wisconsin and chairman of the Homeland Security and Governmental Affairs Committee.

“Time out,” Mr. Johnson added. “We wouldn’t even be discussing this if President Obama hadn’t done what he did.”

What the president did — declaring through his powers as the chief executive to make up to five million undocumented immigrants eligible for new legal protections and rights, including work permits — has left Republicans divided over how to fight back.

The party has struggled to come up with an alternative to the current Homeland Security bill, which includes provisions approved by the House that would rescind those new legal protections and subject many people here illegally, including children, to deportation.

Even some Republicans have acknowledged that this legislation will never pass the Senate because members in both parties believe it is too harsh. Some, like Senator Susan Collins of Maine, have tried to come up with a more palatable alternative; so far, Ms. Collins has yet to see a vote on her approach.

But conservative lawmakers and the powerful class of right-leaning news media commentators that can heavily influence public opinion expect Republicans to push Mr. Obama harder — even if he is certain to veto legislation that tries to undermine his immigration policies.