Later Tuesday, the Senate will vote on whether to proceed to the Border Security, Economic Opportunity, and Immigration Modernization Act. Cornyn said he would support that motion, but hopes changes are made to the bill.

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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.

"This bill will make our country safer, and I believe it will make our country stronger," Sen. Chris CoonsChristopher (Chris) Andrew CoonsHillicon Valley: Trump hits China with massive tech tariffs | Facebook meets with GOP leaders over bias allegations | Judge sends Manafort to jail ahead of trial | AT&T completes Time Warner purchase Dem senator: Trump Jr. may have given 'false testimony' about meeting with foreign nationals Overnight Finance: House sends Dodd-Frank rollbacks to Trump | What's in the bill | Trump says there is 'no deal' to help ZTE | Panel approves bill to toughen foreign investment reviews MORE (D-Del.) said after Cornyn spoke Tuesday. "[And] it strikes the right balance of those who are here undocumented to come out of the shadows."

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

The bill makes permanent legal residence contingent the Department of Homeland Security (DHS) having 100 percent situational awareness at every segment of the southern border and a 90 percent apprehension rate. Cornyn said he has an amendment — the Results amendment — that agrees with the Gang of Eight border security standards, but would guarantee the standards are met before amnesty is granted to anyone.

“I wonder why it is that they can’t take yes for an answer. If we agree on the standards they set, why won’t they agree to measures of implementation to meet those standards set?” Cornyn said. “We agree on all of these realistic goals. The difference again is my amendment guarantees results, while the Gang of Eight’s does not.”

Some GOP senators have suggested that the bill gives DHS too much say over whether the border is secure, leaving Congress powerless to stop the amnesty program if security measures aren’t met.

Sens. Cornyn and Chuck GrassleyCharles (Chuck) Ernest GrassleyJustice IG says report doesn’t assess ‘credibility’ of Russian probe DOJ watchdog probing Comey's memos, will release another report Grassley demands details on Comey's use of personal email MORE (R-Iowa) have said the bill is similar to the last immigration reform measure in 1986 because it “legalizes first and enforces later.”

“In other words the federal government has always said the right things to the American people, but it has never lived up to its promises,” Cornyn said. “This is doable, but we need a leverage to compel the bureaucracy and Congress to get the result the American people want.”

Sens. Marco RubioMarco Antonio RubioHillicon Valley: Supreme Court takes up Apple case | Senate votes to block ZTE deal | Officials testify on Clinton probe report | Russia's threat to undersea cables | Trump tells Pentagon to create 'space force' | FCC begins T-Mobile, Sprint deal review Feehery: Betting on Trump Senate votes to block Trump's ZTE deal MORE (R-Fla.), John McCainJohn Sidney McCainTrump plan to claw back billion in spending in peril McCain calls on Trump to rescind family separation policy: It's 'an affront to the decency of the American people' Senate passes 6B defense bill MORE (R-Ariz.), Lindsey GrahamLindsey Olin GrahamTrump digs in amid uproar on zero tolerance policy Senate passes 6B defense bill Justice IG says report doesn’t assess ‘credibility’ of Russian probe MORE (R-S.C.), Jeff FlakeJeffrey (Jeff) Lane FlakeSenate chaplain offers prayer 'as children are being separated from their parents' Romney backs Laura Bush on border: 'We need a more compassionate answer' Mark Sanford’s troubles did not begin with Trump MORE (R-Ariz.), Charles SchumerCharles (Chuck) Ellis SchumerDems must stop picking foxes to guard the financial hen house Schumer warns 'House moderates' against immigration compromise bill Trump knocks Schumer, touts North Korea summit in early morning tweet MORE (D-N.Y.), Dick DurbinRichard (Dick) Joseph DurbinLive coverage: FBI chief, Justice IG testify on critical report Hugh Hewitt to Trump: 'It is 100 percent wrong to separate border-crossing families' Opioid treatment plans must include a trauma-informed approach MORE (D-Ill.), Robert MenendezRobert (Bob) MenendezSchumer: Obama 'very amenable' to helping Senate Dems in midterms The Hill's Morning Report: Can Trump close the deal with North Korea? Senate must save itself by confirming Mike Pompeo MORE (D-N.J.) and Michael BennetMichael Farrand BennetOvernight Finance: Trump wants Russia back in G-7 | Senators, allies push back | House approves first fiscal 2019 spending bills | Dems want insider trading probe over job tweet Dems want insider trading probe after Trump jobs report tweet Clinton on his reading habits: 'I like a fast ride' MORE (D-Colo.) comprise the Gang of Eight.