Sen. Rand Paul (R-Ky.) said Sunday that the Senate's bipartisan immigration bill does not do enough to address border security, and he does not plan to support it in its current form.

"Without some congressional authority and without border security first, I can't support the final bill," Paul said on CNN's "State of The Union."

Sens. Bob Corker (R-Tenn.) and John Hoeven (R-N.D.) struck a deal over a border security amendment last week that is expected to assuage the concerns of some Republicans, and help the bill win passage in the Senate by a comfortable margin. But the measure was not enough to win over Paul, whose own amendment to allow Congress to review and vote on border security status annually was rejected.

"It should certainly put to rest any issue regarding border security," Corker said of his amendment on CBS's "Face The Nation." The House Senate will begin voting on the amendment on Monday.

But not all Republicans are sold. Sen. Jeff Sessions (R-Ala.), an opponent of the bill, said that even with the amendment, the bill is not acceptable. "This bill grants amnesty first and a near promise of enforcement in the future, even with Corker-Hoeven Amendment," Sessions said on "Face The Nation."

Sens. Charles E. Schumer (D-N.Y.) and Lindsey Graham (R-S.C.), two members of the bipartisan "Gang of Eight" that crafted the sweeping reform bill, said Sunday that they believe the bill is on pace to win about 70 votes later this week. Over the in the GOP-controlled House, the reform effort faces a much less certain outlook. There, Speaker John Boehner (R-Ohio) has vowed not to bring to a vote an immigration measure that does not have the support of most Republicans. Schumer said winning wide support in the Senate should boost the chances the House will support it.

"I think there will be pressure on Boehner to bring a bill similar to the Senate bill [to the floor]," Schumer said.

Schumer added that Boehner will face a huge backlash if he scuttles the reform effort.

"This has the potential of becoming the next major civil rights movement. I could envision in the late summer or early fall if Boehner tries to bottle the bill up or puts something in without a path to citizenship. ... I could see a million people on the Mall in Washington on the platform, would not be the usual suspects, but the leaders of business, the leaders of the evangelical movement, the leaders of high-tech, as well as most Americans, pressuring the House to act," Schumer said on CNN's "State of The Union."