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It’s unclear how many people the president has already placed out of reach of deportation.

The youth exemption alone has been granted to more than 450,000 people as of Aug. 31, which is the most recent data available. U.S. Citizenship and Immigration Services had been updating those numbers monthly, but it has been more than two months since the last figures were released.

Sen. Jeff Sessions, Alabama Republican, said the GOP shouldn’t accept any immigration bill until Mr. Obama rescinds his nondeportation policies and lets immigration agents do their jobs.

“During his time in office, the president has systematically dismantled interior enforcement, handcuffing immigration officers and bypassing Congress,” Mr. Sessions said. “No agreement should be entered into while such lawlessness continues.”

Both Mr. Obama and congressional Republicans are facing increasing heat from activists. Monday’s detention center blockade was the latest in a string that has included several locations in Arizona, as well as New Orleans, Atlanta and Chicago.

Meanwhile, protesters have staked out House Speaker John A. Boehner’s house and, last week, some were arrested outside the office of House Majority Leader Eric Cantor.

Not every protest is as confrontational. Activists are maintaining an ongoing prayer fast at a tent on the Mall in the District — an event that drew a nod of approval from Mr. Obama in his speech Monday.

The president also repeated his new stance that he is willing to accept an immigration deal from Congress that comes in pieces, rather than the broad single bill that passed in the Senate.

“It’s Thanksgiving — we can carve that bird into multiple pieces,” Mr. Obama said. “But as long as all the pieces get done, soon, and we actually deliver on the core values we’ve been talking about for so long, I think everybody is fine with it. They’re not worried about the procedures, they just want the result.”

Mr. Obama says any final deal must include a pathway to citizenship for illegal immigrants, along with whatever new border security and interior enforcement measures Congress wants.

Many Republicans, though, argue that border security should come first, and legalization later. Many of those Republicans also object to a full special pathway to citizenship, saying that amounts to amnesty.

Instead of a single bill, House Republicans are working on a series of bills, and Mr. Boehner has said he will not enter into negotiations with the Senate on its massive legislation.

Mr. Boehner, though, also said last week that he remains committed to getting something done on immigration, and doesn’t consider it a dead issue in this Congress.

The activists conducting the prayer fast on the Mall urged Mr. Boehner to put that vow into action.

“It’s time Speaker Boehner and the House of Representatives address America’s moral crisis — a dysfunctional immigration system that has continuously undermined the integrity of our national values and the unity of our families,” the activists said in a statement.