With the Senate landmark immigration bill set to go to the full chamber floor for a vote, attention is now shifting to expected heated debate over the version in the U.S. House of Representatives.

Senate leaders received praise for coming to a bipartisan consensus on key issues Tuesday night -- although one involved the very controversial decision to withdraw an amendment including same-sex couples -- thus paving the way for a panel to approve the measure 13-5.

But the immigration plan, which is backed by President Obama and many establishment Republicans, appears headed for a showdown with wary House conservatives.

Immigration is one of several key issues that is throwing into sharp relief the divisions among Republicans just when they'd like to show a united front and take full advantage of President Barack Obama's latest political problems.

All political parties have their divisions, of course. It's possible the controversies dogging the White House will play a much bigger role in next year's elections than will Republican disagreements.

Moreover, mainstream and Tea Party Republicans joined forces in 2010 — the last midterm election — to lift the party to huge congressional and gubernatorial victories. It's entirely possible they will do it again next year.

But the rise of nonestablishment Republicans, such as Paul of Kentucky and Sen. Ted Cruz of Texas, underlines the party's continued struggle to resolve basic philosophical differences after losing four of the last six presidential elections.

Obama and other Democrats have long sought a way to bring millions of immigrants living here illegally out of the shadows. The Republican establishment backs the idea in hopes of starting to heal the GOP's poor standing with Hispanic voters, a fast-growing group.

It's unclear whether the Republican-run House will embrace the Senate proposal, which would create a pathway to citizenship for many who entered the country illegally. Chances are slim.

"There's no bill I've seen that I can support," Rep. Kenny Marchant, R-Texas, said in an interview Tuesday. When his constituents hear explanations of the proposed pathway to citizenship, he said, "they omit that paragraph and pencil in 'amnesty.'"

Even if Congress passes an immigration overhaul, opponents' remarks in the upcoming Senate and House debates could offer sound bites and video clips that Democrats can use to depict Republicans as hostile to Hispanics.

Senate Republican Leader Mitch McConnell of Kentucky vowed not to stand in the way of the immigration bill on the floor.

He praised the Senate bipartisan group – called the Gang of Eight, and which includes four Democrats and four Republicans – for their work on the measure.

“The status quo is not good, the current situation is not good,” McConnell said of the current immigration system.

Some conservatives denounced McConnell as a sell-out.

A story in the D.C. insider publication The Hill noted that one conservative website, RedState.com, ran a piece that said: “Schumer’s immigration deform bill might play well with McConnell’s base of donors and establishment consultants who desire to remake the party, but it won’t play well with his constituents.”

Another conservative leader said, according to the publication, said: "It's pretty surprising. Usually McConnell waits until the last minute to take a position on a controversial issue. Now he's out there on the record hoping for amnesty.”

Several immigration advocacy groups on Wednesday were planning "an action day" outside Cruz's Washington D.C. office to call for what they termed an end to the "old GOP era."

"The resounding defeat of Cruz’s amendments to the immigration bill and other GOP-sponsored anti-immigrant amendments show that the glory days for these lawmakers is over," said a statement by the coalition. "[Immigrant] families will continue to speak of the importance of including a right to reunite in the bill as they tell their stories of the emotional and financial hardship caused by the current broken immigration system."

The Associated Press contributed to this report.

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