Landmark immigration legislation is doomed to fail in Congress unless border-security provisions are greatly strengthened, Republican senators bluntly warned Tuesday.

“If in fact the American people can’t trust that the border is controlled, you’re never going to be able to pass this bill,” declared Sen. Tom Coburn of Oklahoma, top Republican on the Senate Homeland Security and Governmental Affairs Committee.

His admonishment, joined by those of other GOP lawmakers, came as both Democratic and Republican senators filed a flurry of amendments ahead of the first votes Thursday in a separate committee on the far-reaching bill to deal with an estimated 11 million immigrants who are in the U.S. illegally and the millions more who might try to enter in the future. Some of the amendments could destroy the legislation’s prospects by upending the carefully crafted deal negotiated over months by four Republican and four Democratic senators, supporters say.

Border security was the major sticking point Tuesday.

“If we’re going to get immigration reform through, if you’re going to get it through the House, we’re going to have to do a whole lot more on what is the definition of a controlled border than what is in this bill,” Coburn said.

Sens. Rand Paul, R-Ky., and Ron Johnson, R-Wis., voiced similar concerns at a hearing to examine border security provisions of the bill. One of the legislation’s authors, Sen. Marco Rubio, R-Fla., has already acknowledged the bill will face a tough road to passage if those elements are not made stronger, and in a statement Tuesday he welcomed possible changes.

“In order for this bill to become law, it will have to be improved to bolster border security and enforcement even further and to limit the federal government’s discretionary power in applying the law. In addition, additional measures will be required to address potential costs to taxpayers,” Rubio said.

Paul, a tea party favorite who has voiced support for comprehensive immigration overhaul, insisted his goal in raising questions about the bill is to make it better so it can pass not just the Democratic-controlled Senate but also the Republican-run House. He denied that he’s out to kill the measure or slow it down.

“I want to be constructive in making the bill strong enough that conservatives, myself included, conservative Republicans in the House, will vote for this, because I think immigration reform is something we should do,” Paul said.

“If it’s not any stronger than this I don’t see it getting through the House.”

Echoing concerns raised by a number of Republicans, Paul said the bill relies too much on setting goals and requiring studies about border security instead of insisting on actual accomplishments. Under the bill, “You have to have a plan to build a fence, but you don’t have to build a fence,” he complained.

Sen. Dick Durbin, D-Ill., an author of the legislation, defended the border security provisions and said that for some Republicans, border security is just their excuse to oppose immigration overhaul legislation.

Durbin said border security is stronger than ever, but nonetheless “we went the extra step in this bill, and they’re saying it’s still not enough. You kind of reach a point where you have to question their commitment to immigration reform.”

The bill would allocate $5.5 billion for border measures aimed at achieving 100 percent surveillance and blocking 90 percent of illegal border crossers and would-be crossers in high-entrance areas.