WASHINGTON – The Senate kept its fragile immigration overhaul alive Tuesday, beating back a Democratic effort to offer legal status to 1.6 million illegal immigrants who’d be left out under the pending bill.

Senators also endorsed a plan designed to thwart illegal hirings by mandating that all employers check prospective workers’ legal status through a government Web site.

“Without a workable verification system, there’s no point in having a bill dealing with immigration,” Sen. Charles Grassley, R-Iowa, said before his measure was approved, 58-40. Republican Sens. Olympia Snowe and Susan Collins of Maine voted for the measure.

But Sen. John Cornyn, R-Texas, said the employer verification system’s design is badly flawed because it would require checks only for workers hired in the future and would exempt employers from prosecution until the system is proved 99 percent accurate.

“My hope is, we will get serious finally, once and for all, in holding employers accountable, those who cheat and who provide the magnet for so many people to come into this country illegally,” he said.

Racing to complete work on a sweeping immigration overhaul by week’s end, the Senate rejected, 61-37, an amendment by Sen. Dianne Feinstein that aimed at the heart of the carefully crafted compromise.

The California Democrat proposed granting legal status to virtually all 11 million-plus illegal immigrants, placing them on a track to become Americans. That would have overturned the bill’s concept of a three-tiered system that deals with illegal immigrants differently based on their time in the U.S.

Illegal immigrants who arrived after January 2004 would be denied legal status; those here two to five years would have to leave the U.S. to apply to return as guest workers; and those here more than five years could “earn” legalization if they work for six years and pay back taxes and fines.

Feinstein described the three-tiered plan as unworkable, particularly the requirement that 1.6 million illegal immigrants be found and deported.

“There is not one person that is going to go home because of what we do in a bill,” she said. “They are going to stay. … Their lifestyle is going to be clandestine.”

While agreeing that the three-tiered system could prove hard to implement, the legislation’s key sponsors said the Feinstein plan could bring the entire immigration bill down.

“If this amendment were to be adopted, the very delicate, fragile coalition which we have for this bill would, I think, fail,” said Senate Judiciary Committee Chairman Arlen Specter, R-Pa.

The legislation appears on course for approval Thursday, setting the stage for a showdown with the House. Conservatives in the House have voiced strong opposition to what they term amnesty for illegal immigrants and are intent on improving immigration enforcement first.

Republicans, who control the White House and both houses of Congress, are feeling the heat to deliver on an issue that has become a key concern for millions of Americans.

“What all of the Republicans need to understand is that the American people sent us here to solve problems,” Cornyn said. “A nonresponsive answer isn’t going to suffice.”

Congress’ answer is likely to include border security and a temporary worker program for foreigners hoping to come in the future – not legalization for illegal immigrants already here, said Sen. Trent Lott, R-Miss. Dealing with illegal immigrants “is the part that I just don’t think we can come to agreement on, so we shouldn’t worry with it,” he said.

Some on Capitol Hill suggested Tuesday that a leading House conservative, Rep. Mike Pence, R-Ind., had come up with a plan that could bridge the divide between the chambers. The Pence plan would require all illegal immigrants to leave the United States to apply for a six-year guest worker visa.