Those are inauspicious numbers for supporters of the bill, who had hoped its 68-vote bipartisan majority in the upper chamber would increase pressure on House Republicans.

That said, 50 percent of respondents said they would be disappointed if the House did not approve a pathway to citizenship for illegal immigrants, versus four in 10 who say they would be relieved. Some 63 percent of respondents also said the House GOP would be responsible for a failure to get immigration reform passed, with just 20 percent blaming the president.

At an event for Organizing for Action, the pro-Obama advocacy group, on Monday night, House Minority Leader Nancy Pelosi (D-Calif.) said she was weighing a discharge petition that would bring the Senate bill directly to the floor of the House. 

Pelosi would be able to force that vote if she was able to gather the signatures of her entire caucus and 18 Republicans. But the maneuver is seen as a direct rebuke of House leadership, and it would be tough to find that many GOP defectors who both support the bill and would be willing to upset leadership.