So while lawmakers from both parties are privately relying on the White House and its agencies to provide technical information to draft scores of amendments to the immigration bill, few Republicans are willing to admit it. Some are so eager to prove that the White House is not pulling the strings that their aides say the administration is not playing any role at all.

“President Obama’s concept of engaging Congress is giving a speech that nobody up here listens to,” said Alex Conant, a spokesman for Senator Marco Rubio, Republican of Florida, who is an important supporter of the immigration legislation. “If passing legislation is like making sausage, then this White House is like a bunch of vegetarians.”

As senators near a final tally on the 867-page bill before the July 4 holiday, immigration supporters acknowledge serious risks in Mr. Obama’s approach: leaving the public advocacy for a major piece of his legacy in the hands of others. If the bill fails to become law, Mr. Obama will be open to criticism from Hispanics that he did not put the weight of his office behind the legislation.

But Mr. Obama has made some careful public efforts, including a speech last week at the White House in which he strongly endorsed the legislation. On Tuesday while on Air Force One in Europe, he called a Democratic negotiator, Senator Charles E. Schumer of New York, to reinforce his opposition to part of a Republican amendment that would have what the administration views as unrealistically tough requirements for border security.

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.