Senate Minority Whip John Cornyn believes his proposed amendment to the "Gang of Eight" immigration reform package could put the legislation on track to clear the Senate and improve its prospects in the House.

The Texas Republican confirmed that approval of his amendment -- described as a set of sweeping changes to the border security components of the Senate immigration reform bill -- would secure his vote, as long as the existing legislation wasn't weakened in other ways. Cornyn presented his proposal to Republicans during Wednesday's weekly lunch of the conservative Steering Committee, and the senator said the reception it received was "generally positive."

"I think this is the antidote, and not a poison pill, because you're not going to get broad bipartisan support for a bill, I believe in the Senate, much less the House, unless you deal with these issues," Cornyn said in a brief interview.

As first reported by The Washington Post's Jennifer Rubin, Cornyn's "RESULTS" amendment is designed to address several of the Republicans' concerns about the "Gang of Eight" immigration bill already approved by the Senate Judiciary Committee. In particular, GOP senators have expressed reservations about giving the executive branch too much discretion to measure border security and want Congress to have a greater role in determining how the government would judge the border secure - or not.

Cornyn's decision to propose an amendment intended to strengthen the immigration bill politically is not insignificant. The Texan is the No. 2 Republican in the Senate and is up for re-election next year in a state where GOP primary voters can be influential. And, Cornyn's move comes as Sen. Ted Cruz, R-Texas, a darling of the right, continues to voice reservations about the Senate bill's path to citizenship for illegal immigrants, which is key for Democrats but rankles some conservatives.

Cornyn said he has discussed his amendment with Sen. Marco Rubio, R-Fla., a member of the "Gang of Eight" who signaled in a Tuesday evening radio interview that his crucial vote could rest on whether Republican concerns about border security are addressed. But Cornyn's proposal might also receive a boost from reports, confirmed by a GOP source, that the bipartisan House immigration working group negotiations toward a comprehensive immigration bill are on the brink of collapse.

Cornyn said he wrote his amendment with an eye toward giving the Senate bill a fighting chance in the House, where the majority Republicans are surely going to demand tougher border security measures than those currently found in the "Gang of Eight" proposal. However, Cornyn emphasized, he does not believe that his proposal is at odds with the Senate package.

"This does not undermine the structure of what the 'Gang of Eight' has done," he said, adding: "The House is going to be a challenge and if we address a number of their concerns in this bill in the Senate -- we get a strong bipartisan vote in the Senate -- then I think it makes success and the bill on the president's desk more likely."