The amendment, from Republican Sens. Bob Corker of Tennessee and John Hoeven of North Dakota, cleared a procedural vote in the Senate 67-27. Fifteen Republicans voted in favor of cloture, and zero Democrats broke from their party to oppose the measure.

The overwhelming vote Monday accelerates the Senate momentum for immigration reform as its supporters race toward a self-imposed July 4 deadline to pass the bill. The Gang of Eight authors have inched closer to a supermajority of votes that could be as high as 70 after the group struck a deal with Corker and Hoeven that bolsters security resources for the U.S.-Mexico boundary.

( PHOTOS: 20 quotes on immigration reform)

But frustration from some Senate Republicans over what they see as a plodding floor process boiled over Monday, as 14 GOP senators vented in a letter to Senate Majority Leader Harry Reid (D-Nev.) that their amendments weren’t getting heard. They are angry that there haven’t been enough votes on the Senate floor and many of the changes to the bill are being incorporated in a single Corker-Hoeven amendment.

The broad Corker-Hoeven compromise is likely the last major change to the immigration bill. Reid filed cloture on the overall Gang of Eight bill Monday afternoon, which sets up another procedural vote for Wednesday.

Though some senators are demanding more time and opportunities to offer amendments, an agreement to vote on them would require unanimous consent, which appears unlikely.

( WATCH: This week in Congress)

“We could have had three genuine weeks on this bill, processing amendments and having votes,” Iowa Sen. Chuck Grassley, the top Republican on the Senate Judiciary Committee, said during a floor speech Monday. “Yet, we’re forced to vote on packages that were concocted behind closed doors.”

A spokesman for Reid rebuffed the complaints from Republican senators, saying the same senators had blocked agreement on amendment votes throughout the month.

Added Corker: “The very people who are calling for more amendments are those who have objected to amendments being voted on. I think it’s ironic.”

But one by one on Monday, some undeclared Republicans announced that they won’t be backing the Corker-Hoeven compromise. That decision signals they will oppose the underlying legislation when it comes up for a final vote later this week.

( Also on POLITICO: ‘Gang’ warfare over Senate immigration bill)

Sen. John Cornyn (R-Texas) said he couldn’t support a proposal that was “cobbled together” at the last minute. Sen. Mike Johanns (R-Neb.), a retiring senator once eyed as a potential Gang supporter, announced his opposition with a list of eight complaints about the Corker-Hoeven deal.

And Sen. Richard Burr (R-N.C.) wrote on Twitter: “Any effort to deal w/ illegal immigration must secure the border — Corker-Hoeven amendment and current bill do not achieve this goal.”

Corker and Hoeven took to the floor to beat back criticisms from fellow Republicans that their amendment was too long (only 119 new pages, they said), too complicated (a middle-school student in Tennessee could read it in “30 to 45 minutes,” Corker argued) and still too weak on border security.

“People are certainly entitled to their own opinions, and we respect all opinions,” Hoeven said. “But they’re not entitled to their own facts.”

Fifteen Senate Republicans ultimately voted to move forward on the Corker-Hoeven amendment: Corker, Hoeven, Lamar Alexander of Tennessee, Kelly Ayotte of New Hampshire, Jeff Chiesa of New Jersey, Susan Collins of Maine, Jeff Flake of Arizona, Lindsey Graham of South Carolina, Orrin Hatch of Utah, Dean Heller of Nevada, Mark Kirk of Illinois, John McCain of Arizona, Lisa Murkowski of Alaska, Marco Rubio of Florida, and Roger Wicker of Mississippi.

Hoeven indicated that backers of the proposal could pick up more Republican support if they are able to get their amendments heard — such as Sen. Rob Portman (R-Ohio). Portman voted against cloture, saying he was “desperate” to get a vote on his E-Verify amendment. Hoeven added that negotiators were also talking with Sen. Saxby Chambliss (R-Ga.) on getting his support.

The Corker-Hoeven amendment includes a set of five security “triggers” or benchmarks that must be met before current undocumented immigrants who qualify are able to obtain green cards — a process that won’t happen for at least a decade.

Among the conditions that must be met: the government must deploy an additional 20,000 border patrol agents, at least 700 miles of fencing along the border must be constructed, an E-Verify system must be installed nationwide, and an electronic entry-exit system must be set up at air and sea ports.

Corker and Hoeven’s agreement also includes several changes to the Gang of Eight bill meant to entice undecided senators, such as some benefits provisions for Sen. Orrin Hatch (R-Utah) and protections for the Alaska seafood industry – secured by the state’s two senators, Democrat Mark Begich and Murkowski. Final approval of those revisions is slated for later this week.

The Congressional Budget Office said Monday that the Corker-Hoeven proposal will stem illegal immigration by a “greater” amount than the current bill, but couldn’t give a precise figure. In its formal assessment of the Gang bill last week, the CBO said the legislation would stop only 25 percent of illegal immigration into the U.S.

And the border security additions from Corker and Hoeven will cost an additional $38 billion, the CBO said — still cutting hundreds of billions of dollars from the federal deficit over two decades.

Burgess Everett contributed to this report.