The Senate takes up Monday a complex immigration reform bill expected to receive criticism from both sides of the aisle but which several lawmakers predict is the best chance to get legislation passed to address the millions of illegal immigrants in the United States.

The centerpiece of the new legislation is a "Z visa," to be offered to some 12 million illegals if they pay fines, learn English and return to their countries to file paperwork. That would set them on their way toward permanent residency.

Separately, a temporary guest worker program would be set up to allow foreign nationals to work in the U.S. for up to three two-years stints provided they return to their home country for a year in between each work period.

The White House hopes to get the bill through the Senate by Memorial Day, but that may not happen though a test vote is scheduled for late Monday.

Sen. Jeff Sessions, R-Ala., threatening a filibuster, said it is "unthinkable" that the Senate would complete the measure this week.

"I'm prepared to use whatever tactics are appropriate to resist that," Sessions said.

Another Republican lawmaker, Sen. Kay Bailey Hutchison of Texas, was involved in the negotiations with the White House, but said she has "great concerns about the bill" and plans to amend the legislation to mandate that illegal immigrants return to their home countries before gaining legal status.

"I do have concerns that people can come here and not ever have to go back and get right with the system," Hutchison told FOX News.

Even before the scheduled debate began, the bill sparked heated exchanges on Capitol Hill.

Two Republican senators — John McCain of Arizona and John Cornyn of Texas — had a fiery behind-the-scenes, sometimes profane, exchange last Friday in the vice president's ceremonial room near the Senate chambers over disagreements of the bill, according to reports.

Click here to read about the shouting match in The Washington Post.

Tempers flared, but Cornyn, who opposes the compromise deal, told FOX News on Monday that he accepted an apology from McCain.

"I don't think anything was said that people in that room hadn't heard in some other time before in their lives," Cornyn said. "It was pretty clear that he (McCain) wanted us to move forward so that they could announce their agreement."

Cornyn, who withheld his support for the agreement before last week's public announcement, said he is concerned how the bill will be enforced.

"I think we all feel very strongly that it’s important for us to pass a good immigration bill. There’s just a lot of differences between us as to what that will look like, and that’s what this debate’s gotta be about," Cornyn said.

Click here for more coverage in FOXNews.com's Immigration Center.

But several influential senators from both parties insist the bill does not offer amnesty and is the best chance at compromise.

"To my colleagues who come on the floor to tear this bill down with no alternative, you're not doing this country a service, and I will push back," said Sen. Lindsey Graham, R-S.C., one of the lawmakers involved in the negotiations.

"It is not amnesty, it is well balanced," said Sen. Dianne Feinstein, D-Calif. "It makes major reforms in immigration, it does provide a path for legalization for the 10-12 million."

Senate Minority Leader Mitch McConnell, R-Ky., said the only way to reach a compromise is to move forward with the debate and then take a couple weeks "to give everybody the opportunity in the Senate to feel like they've had their chance to offer amendments that they think would improve the bill."

McConnell said much of the bill's survival depends on a provision that sets triggers to be met on border security and worker documentation before moving forward on giving legal standing to illegals.

"One thing is for sure: If this bill gives them any preferential treatment toward citizenship over people who came into the country in the proper way, that's a non-starter," McConnell said.

On the other side of the aisle, the concern is for preserving families. The bill makes it harder for illegal immigrants to bring over adult children, siblings and parents. Another major objection is the lack of a path to citizenship for temporary workers who are expected to come and go ever few years.

"They're going to bring wage rates down, and after their time is up, they're probably going to stay in the country and become a new group of illegal immigrants," Sen. Charles Schumer, D-N.Y., said.

Looking for a crowning achievement to cap a difficult second term, President Bush said in his radio address on Saturday that the bill brings the U.S. closer to an enforceable border and immigration system.

"I realize that many hold strong convictions on this issue, and reaching an agreement was not easy," Bush said. "I appreciate the effort of senators who came together to craft this important legislation. This bill brings us closer to an immigration system that enforces our laws and upholds the great American tradition of welcoming those who share our values and our love of freedom."

On the House side, conservative lawmakers prepare for their own debate once the Senate passes its legislation.

Rep. Dan Lungren, R-Calif., introduced legislation he said was "an alternative to several of the large holes in the so-called Senate compromise."

FOX News' Malini Bawa and The Associated Press contributed to this report.