Two amendments — one from the left and one from the right — may be the ultimate test of whether a fragile coalition will hold when the Senate brings back the stalled immigration bill.

Supporters of the bill (S 1348) had reason to be optimistic last week when Senate leaders agreed to revive it and limit amendments. The legislation could be on the floor by the end of this week, with 12 Republican and 11 Democratic amendments to be considered.

One Democratic amendment, which would give family ties more weight in any point system that allows illegal immigrants to gain legal status, could cause trouble with Republican supporters of the bill.

The proposal, by Robert Menendez, D-N.J., has support from liberals and immigration advocates. But it has the potential to increase legal immigration and could be a deal-breaker for Republican supporters pushing for a merit-based immigration system.

Menendez dismissed the criticism, saying family reunification is a “bedrock principle” of the nation’s immigration system.

“Some of my colleagues have attached erroneous labels to my other family reunification amendments to rile opposition, but that’s just political rhetoric,” Menendez said.

At the other end of the political spectrum is an amendment by Kay Bailey Hutchison, a Republican who represents Texas, that would require holders of a new “Z” visa — which would be given to illegal immigrants — to return to their home country within two years and reapply for permanent residence.

Either amendment, if adopted, would drastically change the scope of the legislation and potentially derail it again. Senate leadership aides cautioned that the amendment list had not been finalized, but Menendez’s and Hutchison’s offices say they have been assured that their measures will be considered.

Backers of Hutchison’s amendment say her measure would help alleviate conservatives’ concerns about the bill’s legalization provisions.

“This goes a long way toward removing amnesty,” Hutchison spokesman Matt Mackowiak said. “They would have to leave within two years of getting their secure ID. . . . It’s a way to cool down the temperature” on the bill.

Hutchison might get some support from a handful of moderate Democrats who voted against cloture on June 7, but it is not clear whether her measure has enough support to be adopted.

Menendez’s amendment has the support of immigration groups arguing that family unity should be a significant factor in deciding who is allowed to come into the country.

The amendment would allow direct family ties to count for up to 15 points in a 105-point system in determining who gains legal status, up from 10 in a 100-point system in the underlying bill.

Republicans have backed a merit-based system because they believe it would reduce the “chain migration” policies that allow multiple relatives of immigrants to enter the country.

The Menendez amendment “undermines the whole point of moving from a family unification system to a merit-based system,” said one GOP aide working on the bill.

Senate leaders are considering a rarely executed parliamentary maneuver to expedite passage of the immigration measure. The procedural tool, known as a “clay pigeon,” would allow Majority Leader Harry Reid, D-Nev., to offer a single amendment with numerous divisible components. Each component would be one of the amendments that negotiators have agreed to consider.

The approach ensures that each amendment would get a vote at the end of the 30 hours of post-cloture debate, according to Senate aides familiar with parliamentary procedure.

Without the clay pigeon, one of the aides said, the failure to dispose of an amendment by passage or tabling could prevent Reid from calling up and getting votes on the rest of the amendments. That would threaten the carefully negotiated deal. But under Senate rules, the component amendments are all subject to a vote once divided.

Under one scenario, Reid could bring a new version of the bill to the floor, introduce the clay pigeon amendment, have the amendment divided and then hold a cloture vote on the bill before debate and votes on the pieces of the clay pigeon.

Jim Manley, a spokesman for Reid, declined to discuss procedural strategy before the bill hits the floor.

A clay pigeon amendment was last deployed by Tom Coburn, R-Okla., in April 2006 during debate on a supplemental spending measure (PL 109-234). In that instance, Coburn created an amendment with 19 divisible parts, each of which required a vote to strip an earmark.

Republican opponents of the immigration bill say that leadership’s use of such a tactic, usually reserved for mavericks, to silence dissenters could set a bad precedent, and it only fueled their desire to stop the bill.

Jim DeMint, R-S.C., and Jeff Sessions, R-Ala., “have been very upfront in saying they’ll do everything they can to fight this bill,” said DeMint spokesman Wesley Denton.

Discussion of the strategy was in the early stages June 15, and Senate aides cautioned that no final decisions had been made on how to proceed.

Whatever floor strategy Reid pursues, he made clear that it might be a long week in the Senate. Reid said he would keep the Senate in session through the June 23-24 weekend in order to finish the pending energy bill (HR 6) so the chamber can turn to immigration.

“I have insisted on a process to prevent endless debate and get the [immigration] bill through the Senate,” Reid said. “We hope the president will work with us and that congressional Republicans will not stall our efforts.”