Story Highlights• Senate Minority Leader Mitch McConnell says bill still may not pass
• Bill originally yanked off Senate floor June 7 for lack of support
• Senate leaders agreed to bring it back after reaching a deal Thursday
• McConnell says Senate will finish on bill "one way or the other" by July 4
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WASHINGTON (CNN) -- The Senate should wrap up work on a sweeping overhaul of U.S. immigration laws before July 4, but its odds of passage remain uncertain, Senate Minority Leader Mitch McConnell said Sunday.
"It's hard to know whether the votes will be there to pass it or not," McConnell said on 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-Nevada, 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 and bring the bill to a vote. Many said the amendments being considered at the time were nibbling away at the bill's delicate political balance and core aims.
But Senate leaders agreed to bring the measure back after reaching a deal Thursday to consider about 20 amendments, split evenly between Democratic and Republican proposals.
Bush has made reforming U.S. immigration law a priority for his second term, and Thursday's breakthrough came after he backed an amendment to use $4.4 billion in fees raised by the legislation to boost border security and prevent illegal immigrants from being hired in workplaces. (Watch who wants to pass the immigration bill )
The proposal was an effort to woo critics who say the bill needs to place more emphasis on border security.
The bill 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.
The bill's most controversial measure 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." (Watch debate over 'earned citizenship' )
The plan has its critics in both houses of Congress, and Rep. Duncan Hunter, R-California, 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-South Carolina, 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."
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