"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.