We've got to find some way to deal with them (immigrants)," Republican Senator Arlen Specter, the committee's chairman, told ABC television's "This Week" program on Sunday, March 26.
"If they're prepared to work to become American citizens in the long line traditionally of immigrants who have helped make this country, we can have both a nation of laws and a welcoming nation of workers who do some very, very important jobs for our economy."
A bill, passed by the House of Representatives last year, would make it a felony to be in the United States without proper papers, and a federal crime to aid illegal immigrants.
It also allows the construction of a 700-mile (1,126-kilometer) wall along much of the US-Mexico border.
US President George W. Bush, who had made immigration one of his priorities before the 9/11 attacks, has called for "a legal way to match willing foreign workers with willing American employers."
He plans to regularize the status of illegal workers who "fill jobs that Americans will not do."
Since the 9/11 attacks calls for tougher border security have dominated debate over the knotty problem of controlling immigration.
Whether or not the committee produces a bill, Majority Leader Bill Frist plans to open two weeks of Senate debate on the issue Tuesday, March 28. On Sunday, Republican lawmakers grappled over the legal fate of undocumented immigrants in the country, many of whom crossed the southern US border with Mexico.
They also remain divided over how to tackle immigration reform.
Lawmakers have been divided on whether illegal immigrants should be required to return to their home country before they become eligible for US citizenship.
Frist has offered a measure that would punish employers who hire illegal immigrants and provide more visas.
It sidesteps the issue of whether to let illegal immigrants already stay in the United States.
But employers and immigration advocates prefer a bill drafted by Sens. John McCain and Edward M. Kennedy that would allow illegal immigrants to become eligible for permanent residency after working for six years.
Another approach offered by Sen. John Cornyn and Jon Kyl would let illegal immigrants get temporary work permits for up to five years. They would have to leave the United States but could then apply for legal re-entry.
Protests are planned before the US Capital in Washington Monday to protest against tightening immigration laws in a country that is home to some 11 million undocumented immigrants. Members of the clergy plan to wear handcuffs to protest what they said is the criminalization of their aid programs for poor immigrants.
One of the most powerful institutions behind the wave of public protests has been the Roman Catholic Church, the Washington Post reported Monday.
In recent weeks, the church has unleashed an army of priests and parishioners to push for the legalization of the nation's illegal immigrants, sending thousands of postcards to members of Congress and thousands of parishioners into the streets.
Rallies in support of immigrants nationwide have astonished even organizers.
"It's unbelievable," Partha Banerjee, director of the New Jersey Immigration Policy Network, told the daily.
"People are joining in so spontaneously, it's almost like the immigrants have risen. I would call it a civil rights movement reborn in this country."
Nearly 5,000 protesters marched through the streets of Los Angeles Sunday to protest the legislation, one day after a record of half-a-million people demanded amnesty for the undocumented immigrants.
Also Saturday, some 100,000 people marched in Chicago, 30,000 in Milwaukee, Wisconsin, and 15,000 in Phoenix, Arizona, against the bill.
Rights advocates maintain that the proposed bill paid no heed to the fate of millions of immigrants in the country. "There are 11 million people living and working in the United States. This bill ultimately does nothing about that," said Cecilia Munoz, vice president of the National Council of La Raza, an advocacy group for America's Latino community.
"We really need to deal with that reality if we're going to have a law that's effective and that works," she told NBC.
Business owners also complain that the bill harms their work, asserting it is already hard for them to persuade Americans to perform the unskilled jobs that immigrants easily fill, according to the Washington Post.
"I don't think it's a wage situation. It's the type of work and the nature of the work. It's hard, backbreaking work," said Bill Trimmer of the Professional Grounds Inc., which runs a help-wanted ad to find landscapers and groundskeepers.
"I think we're a more affluent society now. They expect more. Everybody expects more. . . . I have contracts, and they want an affordable price, too."