WASHINGTON (CNN) - The Senate should wrap up work on a sweeping overhaul of U.S. immigration laws before the Fourth of July, but its odds of passage remain uncertain, the chamber's Republican leader said Sunday.
Despite an agreement last week that revived the White House-backed measure, "It's hard to know whether the votes will be there to pass it or not," Minority Leader Mitch McConnell told CBS' "Face the Nation."
McConnell predicted the Senate would wrap up work on the measure "one way or the other" before the holiday.
Senate Majority Leader Harry Reid, D-Nev., yanked the bill off the floor June 7 after supporters failed to muster the votes necessary to cut off debate on a series of amendments that were nibbling away at the delicate political balance the bill's authors had crafted. But Senate leaders agreed to bring the measure back after reaching a deal to consider about 20 amendments, split evenly between Democratic and Republican proposals.
The measure would create a guest-worker program to let migrant workers from other countries work temporarily in the United States, a plan that critics have said would create a permanent underclass of poor, low-skill workers.
But the bill's most controversial aspect is the creation of a pathway to legal status and eventual citizenship for the estimated 12 million illegal immigrants already in the country - a plan critics denounce as "amnesty."
Bush has made reforming U.S. immigration law a top priority for his second term, and Thursday's breakthrough came after he backed an amendment that would that would use $4.4 billion in fees raised by the legislation to boost border security and prevent illegal immigrants from being hired in workplaces.
The bill drew support from just seven Republicans on the procedural motion that led to it being shelved, along with 37 Democrats and one independent - a total of 15 votes shy of the 60 needed to move forward. Reid said Bush needed to bring more GOP allies around to supporting the bill, and Bush's proposal was an effort to woo critics who say the bill needs to place more emphasis on border security.
But the plan still has its critics in both houses of Congress, and Rep. Duncan Hunter, R-Calif., told CNN's "Late Edition with Wolf Blitzer" that the Bush-backed amendment is "a terrible trade."
"Border security is the obligation of the American government," said Hunter, a GOP presidential candidate. "That's like saying we'll send enough bullets to our troops in the field in Iraq or Afghanistan if you do something else, if you in Congress will make the right move. That should be a given."
Sen. John Cornyn, R-Texas, called Bush's plan "a good idea." But he added, "That alone is not going to satisfy the concerns about whether we are really going to build a workable system."
"Part of the problem is the American people look at this and they remember what happened in '86, when they were told, if you'll accept a one-time amnesty, then we'll get true enforcement," he told CNN. "Well, we all know what happened. We got an amnesty, but no enforcement."
But Sen. Lindsey Graham, R-S.C., said it would be "political malpractice" to simply focus on current law without addressing the status of the immigrants now in the United States illegally.
"This is no longer about immigration reform. This is about, can we govern ourselves?" Graham said. "Can Republicans and Democrats sit down at a table and do the hard things, or are those days behind us? I am confident that the Senate will deliver."