"There are 3 million people every year coming across our borders illegally. We don't know who they are; we don't know what their intentions are. We absolutely must address it," said Frist, R-Tenn. "I hope by Friday that we will have a bill on the floor that is comprehensive."
A chief sponsor of a House bill, meanwhile, also called on the Senate to avoid deadlock so lawmakers in both houses can start work on reaching a compromise "for our national security and our economic well-being."
"No bill will end up being the worst of all possible worlds," said Rep. James Sensenbrenner, R-Wis., chairman of the House Judiciary Committee. "This will be tough, and it's the toughest thing that I've done in 37 years in elective public office. But it is an important priority."
"Deportation is unrealistic," Sen. Dick Durbin, D-Ill, told CBS News' The Early Show. "To think that we could remove, charge with a crime and remove 11 or 12 million people is unrealistic. And the concept of amnesty has been rejected by everyone. "
The Senate Judiciary Committee last week approved a bill aimed at strengthening enforcement of U.S. borders, regulating the flow into the country of so-called guest workers, and determining the legal future of the illegal population scattered across all 50 states.
The Senate version, which passed 12-6 in committee and was broadly endorsed by President Bush, goes further than the House bill that imposes criminal penalties, proposes building a fence along the borders and is limited to enforcement.
Still, several lawmakers, including Frist, have criticized as unacceptable the Senate provisions that would let illegal aliens already in the U.S. seek citizenship without returning to their home country, paying fines and learning English.
Saying that issue was the "fundamental question," Frist said Sunday on CNN's "Late Edition" that he believes the final Senate version will address ways to provide eventual legal status to some of the 11 million illegal immigrants in the U.S.
"I don't think we should legislate a track that gives a privileged status to people who broke the law," Frist said. "If somebody is here and they're a felon or multiple misdemeanors or somebody who is not working, someone who has been here for a year ... yes, I think they'd have to go back home."
Sen. Lindsey Graham, R-S.C., predicted the full Senate will approve the committee's bill and doubted that some Republicans will try to stage a filibuster to block it.
"It would be political suicide for our party to filibuster a comprehensive solution to a real problem facing America," Graham said on "Fox News Sunday." "It would be political suicide to ignore there's 11 million people, illegally undocumented, who are trying to work and add value to our country."
Durbin, the Senate's No. 2 Democrat, said it is "conceivable" the Senate will reach consensus but called the House bill unacceptable. Allowing illegal aliens already here to seek citizenship is necessary since deporting millions of workers is unrealistic, he said.
"People will have to demonstrate that they're working hard and they're paying taxes, that they have no criminal record," said Durbin.
"Some will be able to meet these requirements; some will not. But it's the only way to bring people out of the shadows and have a system consistent with American values," he said.