Sen. Lindsey GrahamLindsey Olin GrahamCongress punts fight over Dreamers to March Pence tours Rio Grande between US and Mexico GOP looks for Plan B after failure of immigration measures MORE (R-S.C.) on Sunday said the Senate is "on the verge" of getting 70 votes to pass the Gang of Eight's immigration bill, crediting the recent deal on border security language with winning more GOP votes.

"The bill will pass. I think we're on the verge of getting 70 votes. That is my goal," Graham said on "Fox News Sunday." "We're very, very close to 70 votes. The Hoeven-Corker amendment I think gets us over the top."

The Senate is slated to vote on the border security amendment by Sens. Bob CorkerRobert (Bob) Phillips CorkerCongress punts fight over Dreamers to March Drama surrounding Shulkin — what is the future of VA health care? Blackburn pushes back on potential Corker bid: 'I'm going to win' MORE (R-Tenn.) and John HoevenJohn Henry HoevenSenate Republicans call on Trump to preserve NAFTA GOP anxious with Trump on trade GOP lawmakers to Trump: Don't fire Mueller MORE (R-N.D.) on Monday. The amendment is viewed as key to securing more GOP votes for the bill.

Senate Majority Leader Harry ReidHarry Mason ReidWATCH: There is no Trump-Russia collusion and the media should stop pushing this The demise of debate in Congress ‘North by Northwest,’ the Carter Page remake MORE (D-Nev.) wants to finish work on the sweeping immigration bill this week and hold a vote to end debate on the underlying bill on Thursday.

Graham, a member of the bipartisan Gang of Eight, called the Corker-Hoeven amendment "the most aggressive attempt to control the Southern border and regain our sovereignty."

He stressed that Republicans will be blamed if the bill fails to pass the Senate, which would continue to drive Latinos away from the party. The GOP is seeking to court more Latino voters after an overwhelming majority of them voted for President Obama in the 2012 election.

"If it fails, then we are blamed for its failure," Graham said. "Our party is in trouble with Hispanics, not because we're conservative, but because of the rhetoric and the way that we've handled this issue."

"I want to get reattached to the Hispanic community…pass comprehensive immigration reform, grow this party," he said. "The Hispanic community is very close to our values, but we have driven them away over this issue. Let's fix this problem for the good of the country and for the good of the party, and this bill does that."

Sen. Mike LeeMichael (Mike) Shumway LeeThe 14 GOP senators who voted against Trump’s immigration framework Prison sentencing bill advances over Sessions objections Grassley ‘incensed’ by Sessions criticism of proposed sentencing reform legislation MORE (R-Utah), an opponent of the bill, however, criticized the Gang of Eight measure for wrapping the most "contentious" issues on immigration into a single bill that spans more than 1,000 pages. He praised the House's approach of tackling immigration in a piecemeal fashion.

"What's making this contentious, what's making it politically divisive, is the Senate Gang of Eight has insisted on wrapping all of these issues up into one 1,200 page bill," Lee said also appearing on "Fox News Sunday." "They're putting the single most controversial…issue, that of amnesty and ultimate citizenship, into a 1,200 page bill."

Lee also said the Senate bill lacked teeth on border enforcement and few members have had the chance to fully weigh the merits of the Corker-Hoeven amendment. However, he predicted the bill will ultimately pass the Senate.

"They told us [the bill] would be tough but fair, and it's neither. It's not tough on those who have broken the law and it's not fair for those who have been patiently waiting their turn in line to come to this country legally," Lee said.

"Look, we all want immigration reform, but this bill is not the vehicle for getting there," he said.