He and fellow conservative Rep. Michele Bachmann of Minnesota described the GOP caucus as evenly divided on the question.
"The real issue is amnesty. The whole game is about legalization, the status of legalization," Bachmann said.
On the other side was Rep. Ileana Ros-Lehtinen, a Cuban immigrant from Miami, who told reporters that "we have different points of view."
"I hope we can also look at the 11 million people who want to legalize their status," she added.
For Obama and Democrats, passage of immigration reform would mean following through on a promise to the nation's largest minority demographic -- Hispanic Americans -- to reduce uncertainty for millions of them living illegally in the United States.
Moderate Republicans, including most of the party's leaders in Congress, want a compromise that would represent a potential breakthrough for the GOP with Hispanics who voted overwhelmingly for Obama in last year's election.
Earlier Wednesday, former President George W. Bush added his voice to calls for a solution, telling a ceremony for new citizens that the present system was broken.
"I do hope there is a positive resolution to the debate," Bush said in a rare public comment about politics since he left office. While he refrained from taking sides, he sounded moderate in saying he hoped that "we keep a benevolent spirit in mind."
"We understand the contributions immigrants make to our country," said Bush, a Republican. "We must remember that the vast majority of immigrants are decent people who work hard and support their families and practice their faith and lead responsible lives."
White House keeps up pressure
However, conservatives fear that legalizing undocumented immigrants means guaranteeing future electoral hardship for Republicans by adding millions of new voters who generally lean Democratic.
The White House sought to keep up pressure on House Republicans, asserting Wednesday that the Senate immigration measure would boost the U.S. economy and help create jobs.
A Congressional Budget Office report released in June indicated the Senate bill could reduce deficits by $175 billion over the first 10 years and by at least $700 billion in the second decade. The CBO, working with the congressional Joint Committee on Taxation, estimated that eight million unauthorized residents would become legal in the first decade.
In addition, the CBO report estimated the bill would boost the U.S. population by a net of 10.4 million people by 2023 and by 16 million by 2033.
"The benefits are clear. The fact that there is a broad bipartisan consensus behind this is clear," White House spokesman Jay Carney said Wednesday.
"It cannot be acceptable, broadly and in the long term, that immigration reform would be blocked because some minority of House Republicans is concerned about a primary challenge from the far right. That's not a good argument. It's not a good argument politically. It's certainly not a good argument economically," he said.
On the House side, the bipartisan group working on an immigration package would require that border security measures be in place before any process toward residency for undocumented immigrants could begin.
GOP legislators endorsed that strategy after Wednesday's caucus meeting.
"The American people want our border secured, our laws enforced, and the problems in our immigration system fixed to strengthen our economy," said a joint statement by House GOP leaders that expressed distrust in Obama and Democrats to fully enact tougher security laws before legalizing undocumented immigrants.
GOP Republican Rep. Darrell Issa of California told reporters the end result in the House would be "fully comprehensive immigration reform" that would address undocumented immigrants.
He outlined legislation that would provide some with legal status, such as those who served in the military and children of undocumented immigrants brought into the country illegally by their parents.
While other immigrants also could be considered for legal status, those with criminal records should be kicked out, Issa said.
Sticking points in the House debate include whether undocumented immigrants would be eligible for health care benefits during the years it would take to get legal status, and empowering state and local police to work with federal authorities in enforcing immigration laws.