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A Senate immigration proposal has garnered enough votes to clear a key hurdle, though late-arriving lawmakers continue to vote.

Supporters of the proposal being voted on Monday have attracted at least 66 votes. The measure needs 60 to pass.

Flight delays have kept some lawmakers from the floor, but the Senate is keeping the vote open.

The measure being voted on Monday is an amendment to the main bill, but is considered a key test of whether the bill itself has enough support to pass the full chamber.

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    A top Senate Republican sounded the alarm ahead of a major test vote on Monday that could clear the way for passage of a sweeping immigration overhaul, saying the chamber was about to vote on a bill "no one has read."

    Lawmakers have since begun voting and should wrap up shortly after 6 p.m. ET.

    Sen. Jeff Sessions, R-Ala., who for months has been fighting against the bill, accused the Senate of rushing to vote on the legislation amid a late push to modify it. At issue is a border security amendment being touted by supporters as a bipartisan compromise that could attract a large majority.

    But Sessions accused Senate Majority Leader Harry Reid of cramming the amendment into the rest of the "1,200-page" bill with the goal of advancing the legislation late Monday afternoon.

    "The Majority Leader's motion will stifle amendments and accelerate the vote on final passage before anyone has vetted the modified legislation," Sessions said in a statement.

    Sessions took to the Senate floor on Monday afternoon to complain that the updated language was only filed Friday.

    "This is exactly what happened with ObamaCare," Sessions said.

    Reid, meanwhile, declared that "the immigration bill will pass with Democratic and Republican votes."

    Reid, speaking to Fox News, later said he anticipates the bill will pass with between "68 and 71 votes."

    Tensions were high ahead of the vote, which is technically on the amendment itself. Reid needs to gather at least 60 votes to advance toward final passage, possibly before the holiday recess. Some conservative Republicans acknowledged the bill is likely to pass the Senate, but warned that the House is another matter.

    "It's dead on arrival in the House," Sen. Rand Paul, R-Ky., said on Sunday.

    But the sponsors behind the border security amendment -- Sens. Bob Corker, R-Tenn., and John Hoeven, R-N.D. -- staunchly defended their effort ahead of Monday's vote.

    "I've seen reports of a '1,200 page bill' no one has read or had time to read," Corker said in a statement, presumably in response to Sessions. "To be clear, the tough border and interior enforcement provisions that Sen. Hoeven and I offered on Friday make up 119 pages added to the 1,100 pages that have been public since May."

    His office said the proposal would require an "unprecedented surge of security" on the border.

    President Obama is also hosting a meeting Monday at the White House with eight CEOs, business owners and entrepreneurs to discuss immigration reform, and to push for support of the bill among the business community.

    Obama is expected to emphasize a report released by the Congressional Budget Office last week that said the bill would increase the real GDP by up to 3.3 percent in 2023, and by 5.4 percent in 2033.

    The group of senators that crafted the legislation is trying to get 70 votes to show the bill has widespread bipartisan support in the Democrat-controlled chamber and to give it momentum as it heads into the Republican-controlled House with a more uncertain future.

    Last week, senators proposed the so-called Border Surge amendment, which included 70,000 additional U.S. border agents and 700 more miles of border fencing, to garner support from lawmakers who said the influx of illegal immigrants remains a problem and to put added political pressure on House conservatives.

    But some conservative groups were skeptical. The Heritage Foundation on Monday warned that the amendment would allow illegal immigrants to "receive amnesty now," with the possibility of more border security "somewhere down the road."

    Paul told CNN's "State of the Union" that lawmakers in the House "think border security has to come first before you get immigration reform."
    New York Sen. Chuck Schumer -- a Democratic member of the bipartisan, so-called Gang of Eight that crafted the legislation -- also predicted the bill will get 70 votes and would "change the dynamic in the House."

    Schumer told CNN the bipartisan support for the legislation that should result in the 70 votes also will put "huge pressure" on House Speaker John Boehner, R-Ohio, not to block immigration reform.

    If the bill passed the Senate, Boehner will be faced with honoring the will of the majority of House Republicans who don't appear to want to pass the legislation or honoring the majority of the chamber -- some Republicans and some Democrats -- that appears to want at least a full floor vote.

    He also must consider what message blocking the legislation will send to Hispanic voters, who gave President Obama roughly 70 percent of their vote in the 2012 election.

    Still, Sen. Mike Lee, R-Utah said passing the roughly 1,200-page bill is a mistake. He continues to argue that Congress should take a more step-by-step approach, starting with further securing the U.S.-Mexico border.

    "It could take years to implement the border-security measures," he said.

    Lee said the lawmakers crafted the bill with the "best intentions" but failed.

    "They said it is tough and fair, but it's neither," he said.

    The bill would provide a years-long path to citizenship for the roughly 11 million illegal immigrants now living in the U.S.

    Lee was joined on Fox News by South Carolina Sen. Lindsey Graham, a Republican member of the Gang of Eight.

    "We are very, very close," Graham said. "The amendment gets us over the top."