Reasons for opposing any kind of legalization range from punishing lawbreakers to political protectionism, with conservatives fearing that most immigrants given what they call amnesty and the eventual right to vote will lean Democratic.
However, the issue of legalizing immigrants is broad and complex, creating lots of uncertainty.
For example, the Senate bill would automatically give immigrants living illegally in the United States temporary legal status as "registered provisional immigrants." Only when certain border security steps had been taken could they apply for permanent residency, or green cards, as a step toward potential citizenship in process that would take more than a decade.
Many House Republicans made clear they don't want any kind of legal status for undocumented immigrants until the borders are secure. Even those open to legalization don't want it to include a path to full citizenship.
The labels and definitions of legal status will be a major sticking point in the continuing debate, but also could be a source of compromise.
House Majority Leader Eric Cantor of Virginia told Wednesday's GOP caucus meeting that children of undocumented immigrants brought illegally to America through no fault of their own should be provided a path to legal status, a position strongly favored by Democrats.
The backing of Cantor and other House Republicans for such a provision showed room for maneuvering exists.
After meeting with Obama at the White House on Thursday, GOP Sen. John McCain of Arizona called on House Republicans to negotiate an immigration bill.
"We are ready to sit down with you and negotiate and bring this issue to a conclusion," said McCain, part of the bipartisan Senate "Gang of Eight."
Sweeping reform isn't popular with GOP in either chamber.
There may be more Senate GOP support for comprehensive immigration reform, but not that much. Only 14 of 46 GOP senators backed the "Gang of Eight" bill heralded in its creation as a triumph of bipartisanship in a sharply divided Congress. Why should House Republicans be more in favor?
Remember that all politics is still local -- especially in the House. Many House Republicans represent ruby red districts with few Hispanics, where any path to citizenship is unpopular and the big fear is a primary challenge from the right.
Which leads us to ...
The Hastert rule
House Speaker John Boehner has made clear that the House will only take up immigration reform that is backed by a majority of its Republican members. That is keeping with the maxim of former House Speaker Dennis Hastert that prevented votes on legislation that lacked strong support from the controlling party.
Democrats contend the Senate version would pass the House with a few dozen Republicans joining them to overcome opposition by most of the GOP caucus.
While it is unclear if that's true, permitting it to happen would antagonize many of Boehner's fellow Republicans.
"If the speaker allows a vote on any immigration bill that results in passage despite a majority of the Republican conference voting against it, then it will be interesting to see if he can muster the votes to get re-elected after the next election," Alabama GOP Rep. Mo Brooks recently told CNN.