Senate conservatives have been warned by Republican leaders that they must either accept a series of largely symbolic floor votes on a handful of amendments to the immigration reform legislation or see themselves shut out of the process altogether when the chamber resumes work on the bill later this year, GOP lawmakers and aides said Tuesday.

The warning came as President Bush met privately with Senate Republicans in an effort to restart immigration talks that stalled last week after a failed effort by Senate Majority Leader Harry Reid (D-Nev.) to end debate on the controversial measure.

Although Minority Leader Mitch McConnell (R-Ky.), Minority Whip Trent Lott (R-Miss.) and other leaders voted against several cloture motions late last week, they are pressing their Members to accept a deal under which Republicans could offer about a dozen amendments with a set time for debate.

But leadership also has made it clear that if conservatives do not agree to the deal, they run the risk of losing their seat at the negotiating table.

A senior GOP leadership aide confirmed that conservatives have been given the take-it-or-leave-it deal, saying lawmakers have been told they “need to get on board if they want to remain relevant.”

Sen. John Cornyn (R-Texas) said conservatives are receiving “significant encouragement to get down to a list of 10 or 15 amendments on the Republican side.”

An aide to another conservative lawmaker complained the deal that GOP leadership is forcing on Republicans is unfair. “The implicit threat is that if you don’t go along with this, you won’t get a vote,” the aide said.

But even that offers little comfort to conservatives, since the votes’ outcomes are essentially a foregone conclusion.

Because the bipartisan group of Senators who have agreed to the “Grand Bargain” is expected to continue to vote as a bloc against any amendments that might derail the overall bill, the additional amendments that would be granted floor time are unlikely to pass.

Sen. John Thune (R-S.D.), an opponent of the legislation, said that while having the ability to offer amendments would ease some of his unhappiness with the process, the predetermined fate of the amendments made it a hollow deal at best.

While “a couple of amendments have snuck through … [it has been] a pretty orchestrated process,” Thune said, adding that “when you’ve got a small group screening the amendments, that’s a process that’s a little bit rigged from the start.”

Nevertheless, McConnell is under increasing pressure from many of his members and President Bush to break the deadlock so a bill can be moved out of the Senate.

Sen. Larry Craig (R-Idaho) said he already has told Senate GOP leaders he will not support the immigration filibuster any longer. “They’ve had ample opportunity,” Craig said of the Republican dissenters who successfully rallied a majority of their Conference to block the bill last week. Craig added that he supports McConnell’s efforts to goad the conservatives into capitulating, saying that over the years “when I was in a minority position I was given fair fun, but I was never given carte blanche to kill something.”

Craig said the president seemed receptive to the notion that an emergency supplemental spending bill should be used to fund the border enforcement provisions of the bill.

“Getting it right, in part, is funding what we’ve done,” said Craig, who said Americans are understandably nervous about whether the government actually will enforce immigration laws. “Our great challenge is to prove our credibility,” he said.

Though Craig said it was unlikely Bush’s appeal to Republicans today changed the minds of stalwart opponents, such as Sens. Jim DeMint (S.C.) and Jeff Sessions (Ala.), he said, “the president certainly did not hurt the situation today.”

Democratic aides said on Tuesday that it is unclear when immigration could return to the floor, though one leadership aide said it would not be before the end of the energy debate, which is expected to last two weeks.

Additionally, while conservatives said they believe passage of the bill is now inevitable, it also is unclear whether Reid and McConnell ultimately will be able to come to an agreement on floor time for the measure.

Cornyn noted that Reid already rejected a similar offer of 10 amendments on Thursday before he forced a second vote on bringing debate on the bill to a close.

“So I’m not sure how sincere [Reid’s] statement is that, if he gets that list, he’ll be willing to give those amendments votes and an adequate time to debate them,” Cornyn said.

Emily Pierce contributed to this report.