"Mr. Boehner, you are on the clock," Rep. Luis Gutierrez, D-Ill., said following the Senate vote Thursday.
"Today, the Senate did its job," President Obama said. "It's now up to the House to do the same."
Boehner is caught between the conservative elements of the House's Republican caucus wary of the immigration plan passed by the Senate and the national GOP leaders backing comprehensive reform, if only to improve the party's standing with Hispanic voters, The Hill reported Friday.
Boehner on Thursday declared he won't consider any immigration legislation, including a conference report, unless it has support from a majority of his party. He also restated the Senate-passed bill won't be considered on the House floor.
"We're going to do our own bill through regular order, and it'll be legislation that reflects the will of our majority and the will of the American people," Boehner said during a news briefing in the Capitol.
"The coalition for comprehensive immigration reform? I don't think the House of Representatives quite understands how broad and deep it is, because it's been perpetually stationed in the Senate for the last four months," Gutierrez said before the Senate vote. "Well, they're closing down camp there in about an hour and setting up camp here."
Sen. Charles Schumer, D-N.Y., one of the bipartisan "Group of Eight" that crafted and shepherded the bill, said the 68-32 Senate vote that included support from 14 Republicans "set the tone so Republicans go to Boehner and say, 'Look, I don't want to vote for this, but get it off our back.'"
Another Gang of Eight member, Sen. Jeff Flake, R-Ariz., said he would begin calling members of the House Judiciary panel, The Hill said.
"There is going to be a lot of pressure to take something up," Flake said of House leaders. "It's their turn."
Senate opponents noted that the bill fell short of the 70 votes supporters hoped to pick up, The Hill said.
Boehner said Thursday Republicans hope to use the July 4 recess to meet with constituents, and the entire conference will meet July 10 to discuss the next steps.
"I don't want to make any predictions on what the outcome of that conversation is going to be," Boehner said, "but we're going to have a conversation and determine a pathway forward."