“It’s amazing to me that the Majority Leader has touted this as an open process … but on the first day, they want to hold our amendments to 60-votes," Grassley said. "Who’s obstructing now?”

Reid said he was just following the lead of Republicans, who have repeatedly blocked legislation during the 113th Congress, requiring 60-vote thresholds to advance bills.

“How many times have we heard the Republican Leader [Mitch McConnellMitch McConnellSenate Dem: We’re trying to block a recess appointment to replace Sessions Trump predicts 'problems' for those voting against ObamaCare repeal Overnight Defense: House passes Russia sanctions deal | McCain returns to Senate | Watchdog opens criminal probe into M camo mistake MORE (R-Ky.)] say 60 votes is how things should get done in the Senate,” Reid said. “The McConnell rule is 60 votes on everything.”

Grassley said Reid’s request proves GOP complaints that supporters of the Gang of Eight’s bill really don’t want any changes made.

“It looks like the fix is in, and the bill will have to pass as is,” Grassley said.

On Tuesday, the Senate voted 84-15 to proceed to debating the Border Security, Economic Opportunity, and Immigration Modernization Act. The Senate will continue to work on the bill throughout June, although if progress isn’t made on the amendment process, it could take longer.

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.

Some Republicans have complained that the legislation would provide amnesty over 10-13 years for the nearly 11 million residents in the U.S. illegally before strengthening border enforcement. 

Reid hinted that if an amendment agreement wasn't reached, the majority could start voting to table GOP amendments, which allows a simply majority to kill an amendment.

“I’m sorry that we’ve had this disagreement here,” Reid said. “But I would say to everybody that there are other ways of having simple majority votes and if there is an objection to this we're going to have to go to that.”