Inside Room 201, the administration has gathered a collection of its own Congressional lobbyists, policy specialists and experts from an alphabet soup of the agencies that will have to put the immigration legislation into effect if it passes. They all moved into the vice president’s offices on June 10, setting up laptop computers and thick binders filled with proposed amendments on an oval conference table.

“We have folks who know the Senate really well, who know the players, who have been through this before so they know exactly what Senate staff needs,” Ms. Muñoz said. “We are deeply, deeply engaged.”…

At one point, Mr. Pagano, Ms. Escobar and the other White House advisers huddled for 45 minutes in the smaller of the two rooms with Mr. Leahy’s top aides. Working from spreadsheets, they discussed each of the 10 amendments that Mr. Leahy was likely to bring to the floor for a vote that day.

“When Republican amendments are filed and we are trying to decide, ‘Can we accept this? Can we accept this without some modifications?’ they are the ones who tell us, ‘This is quite doable,’ ” said one Democratic Senate leadership aide, who requested anonymity to talk about legislative strategy.