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Mr. Graham and Mr. McCain also had praise for Sen. Marco Rubio, Florida Republican, who is a member of the negotiating team who’s acted as a bridge to conservatives but also has kept advocates and other lawmakers guessing about whether he’ll ultimately support the bill.

“Marco Rubio has been a game-changer in my party. He will be there only if the Democrats will embrace a guest-worker program and a merit-based immigration system to replace the broken one,” Mr. Graham said on NBC’s “Meet the Press.”

After consideration by the Judiciary Committee, floor action could start in the Senate in May, Mr. Schumer said.

Meanwhile two lawmakers involved in writing a bipartisan immigration bill in the House, Reps. Luis V. Gutierrez, Illinois Democrat, and Mario Diaz-Balart, Florida Republican, sounded optimistic that they, too, would have a deal soon that could be reconciled with the Senate agreement.

“I am very, very optimistic that the House of Representatives is going to have a plan that is going to be able to go to a conference with the Senate in which we’re going to be able to resolve this,” Mr. Gutierrez said Sunday on CNN’s “State of the Union.”

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