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.’ ”