While that question hangs over a Senate debate on border security and immigration, most senators agree on allowing undocumented workers to stay at least temporarily. The fight is over whether they should have to leave three years to six years down the road.
Even senators who oppose providing a path to citizenship to illegal immigrants are willing to grant them temporary legal status as long as they register with the government, pay fines and eventually leave.
"Our first obligation is to bring them out of the shadows, make sure we know who they are, why they're here, make sure we have a name and some kind of identification for them," Senate Majority Leader Bill Frist said in an interview with The Associated Press.
"Then there will be a period of time, whether it's three years or six years ... but they can continue to work here and at that point in time - that's where the debate is - do they have to go home or are they put on some sort of path to citizenship?" Frist said.
As the Senate opened two weeks of debate Wednesday night, Republicans clashed over whether providing a path to legal citizenship would lead to more flouting of U.S. immigration laws.
President Bush reiterated support for a temporary worker program as he took off for a meeting in Cancun, Mexico, with host President Vicente Fox and Canadian Prime Minister Stephen Harper.
Mr. Bush said workers should be given tamper-resistant identity cards and go to the back of the line when they seek citizenship.
"I think it makes sense to have a temporary worker program that says you're not an automatic citizen to help, one, enforce the border, and, two, uphold the decency of America," President Bush said Wednesday.
"We should reject temporary status and required departure because they are bad for business," said Democratic Sen. Edward Kennedy, who has proposed allowing illegal immigrants to earn a path to citizenship through work. "What do we gain if millions of immigrant workers who fuel our economy are required to spend weeks - or years or decades under some plans - waiting outside the United States for permission to continue their work?"
Frist dodged the question of what to do about illegal immigrants in the country in the bill he introduced. But other bills that could be offered as amendments tackle that issue.
On Monday, Sen. Arlen Specter, a Republican from Pennsylvania, shepherded legislation containing the Kennedy proposal through the Senate Judiciary Committee on a 12-6 vote. He insisted the bill is not amnesty because illegal immigrants would have to undergo background checks, pay fines, back taxes and clear other obstacles before getting on the "citizenship track." They wouldn't have to leave the United States.
"If there is a better way to bring these 11 million people forward so that we can identify them, we are open to any suggestions which anyone may have," Specter, the committee's chairman, said.
Sens. John Cornyn, a Texas Republican, and Republican Jon Kyl, of Arizona, rail against what they call "amnesty" but would give illegal immigrants six months to register with the government. Those who do, could stay in the U.S., but only for up to five years. They would have to pay $2,000 fines annually for the privilege. Those who don't could be deported.
The immigrants who register could return as guest workers, but would have to apply for legal permanent residency - a step to citizenship - from their home countries.
Under current law, a person who is in the country illegally for more than 180 days cannot re-enter the U.S. for three years. A person in the country illegally for more than a year cannot re-enter for ten years. Those prohibitions would be waived for immigrants who register with the government under the Cornyn-Kyl plan.
The proposal gives illegal immigrants a chance to "get into compliance with our laws," Cornyn said.
"Illegal immigrants must be required to return home so they go through the same legal channels as everyone else," he said. "Otherwise, even with the minimal penalties in the pending amnesty proposal, illegal immigrants would be rewarded at the expense of those who have followed the law."