On the other hand, a vote for legislation like the Senate bill could hold real peril for House Republicans, whose solidly Republican districts reward politicians who take the most conservative positions on issues. A new poll by National Journal found that nearly half of Republican voters, 49 percent, said a lawmaker who backs legislation offering a pathway to citizenship would lose their support. Thirty percent said it would make no difference. Only 15 percent said such a vote would make them more likely to back their incumbent.

“I think most members look at this with a great deal of trepidation,” Mr. Cole said.

What the House’s methodical approach yields may determine the ultimate fate of immigration legislation in the 113th Congress. If the House can pass its own immigration bill, lawmakers will have a counteroffer to bring to the voters next year — even if the House and Senate bills cannot be reconciled into a final package for President Obama to sign.

That dynamic played out in 2006, when the Senate passed a comprehensive immigration bill, the House passed a measure bolstering border security without offering new paths to legal immigration, and both bills died with the 109th Congress.

But a sizable group of conservatives fear that passing any immigration bill in the House would set up House-Senate negotiations stacked to yield a final deal that would be much closer to the Senate’s plan than the House’s. If that group, in opposition, joins balking Democrats, they could ensure that no conservative immigration bill would pass.

In that case, lawmakers from both parties say pressure from Republicans alarmed about the broad political consequences of inaction could mount on House Republican leaders to move toward the Senate’s position — or even take up the Senate bill without changes. That dynamic has played out twice this year. In January, House Democrats — with a minority of House Republicans — passed a Senate-White House compromise to avert massive tax increases and sudden across-the-board spending cuts, after House Republicans sunk the speaker’s more conservative version of the legislation.

In February, a Republican version of the Violence Against Women Act crashed on the House floor, prompting Republican leaders to allow the House to pass the Senate’s version — again relying on a majority of Democratic votes.