On Monday, 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.) filed cloture on the comprehensive immigration reform bill. Grassley said it’s now “unclear if any more amendments” will be voted on.

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The Senate is considering the Border Security, Economic Opportunity and Immigration Modernization Act. Reid has said he wants work completed on the bill by the July 4 recess.

The bipartisan group of eight senators, known as the Gang of Eight, introduced S. 744, which would create a pathway to citizenship for illegal immigrants already in the country, toughen border security, create a guest-worker program and boost high-skilled immigration.

Republicans, including Grassley, have complained that the bill repeats mistakes made during the last major immigration reform in 1986, when people were given citizenship before border security enforcement measures were in place.

Later Monday, the Senate will vote on whether to end debate on a substitute amendment from Senate Judiciary Committee Chairman Patrick LeahyPatrick Joseph LeahyGrassley, Dems step up battle over judicial nominees Popular bill to fight drug prices left out of budget deal Judiciary Dems want public hearings with Kushner, Trump Jr. MORE (D-Vt.). His amendment included the “border surge” language from GOP 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 (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 (N.D.). 

In the underlying bill, an E-Verify system and entry-exit system at air and sea ports must be in place before anyone is given a green card. The amendment from Corker and Hoeven would also make require construction of 700 miles of Southern border fencing, the purchase of more than $3 billion in new technology for border security and the hiring of 20,000 more border patrol agents mandatory before green cards are issued.

"It’s quite obvious that I’m going to vote against this amendment," Grassley said. "It does little to address the legalization-first philosophy."

Corker said his amendment has “five tangible triggers” that Republicans have demanded, taking power out of the hands of the Department of Homeland Security to waive border security provisions. The trigger must be in effect 10 years after the bill is signed into law.

Adoption of that amendment is viewed as critical to attracting more Republican support for final passage, which the bill’s authors want in order to put more pressure on the House to take up the legislation.

But Grassley said the amendment is still weaker than current law.

“We were told we were going to secure the border. Well, we’re going to secure the border after legalization,” Grassley said. “But legalization is going to take place before any plan is in place.”