A "deal" on border security may get attached to the Senate immigration bill, but not everyone is in love with it.

In fact, some Republican senators have rejected it out of hand.

The Senate is debating the major immigration-reform bill offered up by the bipartisan "Gang of Eight," and while its prospects remain uncertain, GOP Sens. John Hoeven of North Dakota and Bob Corker of Tennessee today proposed a new amendment calling for 700 miles of border fence and a doubling of U.S. border agents from their current staffing level of 21,000, Univision's Jordan Fabian reported.

While the new amendment has been hailed as a "deal" that will bring new votes on board with reform, Sens. Jeff Sessions, R-Ala., and David Vitter, R-La., doused the idea in cold, border-security-first water Tuesday afternoon as Corker and Hoeven were unveiling it.

"I do not think this amendment is going to touch many of the objections that I spoke about," Sessions, R-Ala., told reporters after a news conference hosted by conservative senators and the group Tea Party Patriots.

Conservatives have demanded that citizenship (or green cards) for undocumented immigrants must be conditioned on border-security benchmarks. Without these "triggers," the senators have said, immigration reform is a no-go.

And without those triggers, so is the Corker-Hoeven deal, in their eyes.

"Apparently plan B is to bring up this Corker-Hoeven amendment very, very quickly, and a lot of hoopla about border security, and try to pass it then, with that amendment, over 60 votes," Vitter said. "Again, I think it's important that we look at the details of the amendment. Let's actually read it before we vote on it."

"It doesn't truly fix the problem for two main reasons - first of all, because an immediate amnesty, an immediate legalization, happens first," Vitter said. "Secondly, in the Corker-Hoeven bill, because all they guarantee is inputs, basically spending money, there is no measurement of success, of results, of actually securing the border."