Sessions argued that Reid would be able to block any amendment vote he doesn’t want to take once cloture is invoked. Sessions complained that since the “border surge” amendment was filed Friday afternoon, senators haven’t had enough time to review the changes made in the more than 1,000-page bill.

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“This is exactly what happened with ObamaCare,” Sessions said. “The majority rushed through a complex bill so there would be no time to understand what’s in it.”

Later Monday, the Senate will vote on whether to end debate on a substitute amendment from Senate Judiciary Committee Chairman Patrick LeahyPatrick LeahyDems get it wrong: 'Originalism' is mainstream, even for liberal judges Live coverage: Day three of Supreme Court nominee hearing Dems land few punches on Gorsuch MORE (D-Vt.). His amendment included the “border surge” language from GOP Sens. Bob CorkerBob CorkerSenators introduce new Iran sanctions Senators war over Wall Street during hearing for Trump's SEC pick Rand Paul roils the Senate with NATO blockade MORE (Tenn.) and John HoevenJohn HoevenSenate panel considers how to fund Trump’s T infrastructure package A guide to the committees: Senate GOP senators unveil bill to give Congress control of consumer bureau budget MORE (N.D.), which requires 700 miles of Southern border fencing and hiring 20,000 more border patrol agent among other things before immigrants can apply for green cards.

Corker said his amendment has “five tangible triggers” — which have to be achieved in 10 years — that Republicans have demanded, taking power out of the hands of the Department of Homeland Security to waive border security provisions.

“If you think border security is not OK under the status quo, vote for this amendment,” Corker said. “If you want to give full control to Janet Napolitano, don’t vote for this amendment.”

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.

Corker defended the length of the amendment and bill, saying the amendment is slightly more than 100 pages and the rest of the bill has been available to read for more than a month.

“Any grade-schooler could read this amendment in probably 30 to 40 minutes,” Corker said. “The length issue is something that’s a total myth.”

Another criticism of the amendment is that the increased border security measures come with a price tag of $46 billion, but supporters point out that the Congressional Budget Office (CBO) still estimated a net gain for taxpayers.

“I’ve been in the Senate for six years, and this is the first time that I’ve had the opportunity to vote on a bill that spends $46 billion over 10 years, and, in the same period, returns $197 billion to the Treasury without raising taxes and while promoting economic growth,” Corker said.

The CBO has said in the first 10 years, the bill will generate $197 billion, and in the decade after that, nearly another $700 billion, but Republicans critics of the measure say those gains are to the Social Security Trust Fund and shouldn’t be used to pay for other projects because the money will have to be returned to the trust fund at some point.

"If the money is spent now it won’t be there then and this is how a country goes broke," Sessions said.

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.

There are only a handful of other amendments pending to the bill, although Reid said the bill managers would work with senators to ensure as many votes as possible before the end of the week. Reid has said he still wants to complete work on the bill by the July 4 recess, and filed cloture on the underlying bill Monday morning.