Emilia Hernandez shouts during a rally in downtown Los Angeles on Wednesday. In celebration of May Day, people gathered across the country to rally for various topics including immigration reform. (Nick Ut/AP)

President Obama is warning liberal supporters that their push to make changes in a comprehensive immigration bill could jeopardize the strategy of Senate leaders, who are aiming to win up to 70 votes for the measure.

While much of Washington has focused on objections from Republicans, Obama and other Democrats have mounted a behind-the-scenes campaign in recent days aimed at mollifying advocates, who argue that an 844-page Senate bill excludes too many illegal immigrants and makes it too hard for the rest to become citizens.

The efforts underscore the perilous path ahead for a comprehensive immigration deal, which is one of Obama’s top agenda items for his second term but faces mounting criticism from those on both the left and right.

In a private meeting with a dozen Latino leaders at the White House this week, Obama emphasized that securing a large margin in the Senate is crucial to putting pressure on House Speaker John A. Boehner (R-Ohio) to accept the general framework of the legislation.

The president made clear that he expected the people in the room to support the Senate proposal even if they had doubts about some details, participants said. Once an overarching plan was locked in place by Congress, Obama told the group, the administration would be able to revisit some of their concerns and figure out ways to improve it.

“He said, ‘If the bill were presented on my desk today, I would sign it,’ ” said Janet Murguía, president of the National Council of La Raza, who attended the meeting. “He looked at the advocates and said, ‘We’re not going to get everything we want in this.’ ”

Obama’s strategy represents a calculated bet that throwing his full support behind a bipartisan proposal crafted by four Democrats and four Republicans is the surest way to avoid the pitfalls that have doomed his initiatives on gun control and the budget. As the White House prepares to confront fierce opposition from conservatives, the administration’s ability to galvanize support among liberals represents a tricky, but particularly critical, task.

It won’t be easy. This week, a coalition of Latino and religious leaders held a conference call with reporters highlighting portions of the bill they believe to be too draconian — including high fees and rigorous employment requirements before undocumented workers can gain legal status.

Meanwhile, gay rights advocates said they expect Senate Judiciary Committee Chairman Patrick Leahy (D-Vt.) to introduce an amendment that would provide visas to foreign same-sex spouses of U.S. citizens, a provision that Republicans have said could kill a deal.

At a news conference this week, Obama praised the Senate group for its bipartisan work. He said there were elements of the bill that he does not agree with but emphasized that he supports the overall package: “I do think that it meets the basic criteria that I laid out from the start.”

In private meetings, Obama has told liberal groups that they must be realistic at a time when Republicans control half of Congress.

After the defeat of his gun-control agenda and little progress on a “grand bargain” solution to curbing the national debt, Obama has turned to immigration reform as his best hope for passing a key second-term priority.

“I think the president is worried about his legacy,” Sen. John McCain (R-Ariz.), one of eight co-sponsors of the Senate immigration bill, said during a talk at the University of Southern California this week. “He knows that compromises need to be made.”

The Senate’s proposal faces deep uncertainty in the Republican-controlled House, where conservatives are mounting opposition to the 13-year path to citizenship for undocumented workers. Some Republicans, including House Judiciary Committee Chairman Bob Goodlatte (Va.), have called for votes on smaller, piecemeal bills, a tactic that advocates fear would kill any chance of a sweeping overhaul.

White House officials and their Senate allies hope to win a big enough majority in that chamber to leave Boehner with little choice but to accept a comprehensive framework. Under this scenario, Republican leaders — eager to broaden the party’s appeal to Latino voters in the wake of Obama’s reelection last fall — would feel compelled to put the Senate bill up for a vote, likely passing with support from Democrats and centrist Republicans.

Some senators in the bipartisan group have said they are eyeing as many as 70 votes in support of the proposal, well above the 60 needed to overcome blocking maneuvers. In his meeting with advocates Monday night, Obama did not mention a specific number of votes but said it will be important to maximize support.

“He felt that it was important for us to support the process moving forward,” said Brent Wilkes, executive director of the League of United Latin American Citizens. “There was some concern that there are some folks in our community that will look at provisions and say, ‘This is not fair and this should be better.’ ”

Not all advocates support such a strategy, however. Leahy, whose committee will oversee the “markup” process of adding amendments to the bill, set a Tuesday deadline for amendments and dozens of potential changes are expected to be filed by Democrats and Republicans next week.

Fred Sainz, a spokesman for the Human Rights Campaign, said gay rights groups would not back down from their push to include protections for foreign same-sex spouses, who are barred from obtaining spousal visas under the federal Defense of Marriage Act.

Obama had included such a provision as part of his immigration principles in January, but Democratic lawmakers dropped it from the Senate proposal because of GOP opposition.

Republicans including Sen. Marco Rubio (Fla.), a co-sponsor of the Senate proposal, have warned that a gay rights measure would doom the bill. Rubio also said this week that he expected changes to the legislation, which he predicted would not pass the House in its current form.

Angelica Salas, executive director of the Coalition for Humane Immigrant Rights of Los Angeles, was among a group this week that criticized what members called the excessive hurdles in the bill for undocumented immigrants to gain legal status.

“We understand there are compromises to get to this legislation,” Salas said. “What we’re trying to bring forward to the senators is in their compromises, in the details. We want to make this bill practical.”

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