Updated

Even with one of the Senate’s strongest opponents to the chamber’s sweeping immigration-reform bill saying Sunday that the legislation will likely pass this week with a resounding 70 votes, Kentucky Republican Sen. Rand Paul predicted the legislation was doomed in the more conservative House.

“It’s dead on arrival in the House,” said the Tea Party-backed Paul. “The House is much closer to me.”

Paul made the remarks as Utah Sen. Mike Lee, among the most conservative Senate Republicans, told “Fox News Sunday” the bill is “likely to pass” with as many as 70 votes.

Their remarks comes as the Senate prepares for a preliminary but key vote on the bill Monday night or Tuesday that should result in Senate passage by the Democratic leadership’s goal of July 4.

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% in 2023, and by 5.4% 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.

The Senate last week introduced a 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.

Still, 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, Lee remains steadfast that 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 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.”