WASHINGTON — Heading into a crucial negotiating session on a budget deal on Thursday, President Obama has raised his sights and wants to strike a far-reaching agreement on cutting the federal deficit as Speaker John A. Boehner has signaled new willingness to bargain on revenues.

Mr. Obama, who is to meet at the White House with the bipartisan leadership of Congress in an effort to work out an agreement to raise the federal debt limit, wants to move well beyond the $2 trillion in savings sought in earlier negotiations and seek perhaps twice as much over the next decade, Democratic officials briefed on the negotiations said Wednesday.

The president’s renewed efforts follow what knowledgeable officials said was an overture from Mr. Boehner, who met secretly with Mr. Obama last weekend, to consider as much as $1 trillion in unspecified new revenues as part of an overhaul of tax laws in exchange for an agreement that made substantial spending cuts, including in such social programs as Medicare and Medicaid and Social Security — programs that had been off the table.

The intensifying negotiations between the president and the speaker have Congressional Democrats growing anxious, worried they will be asked to accept a deal that is too heavily tilted toward Republican efforts and produces too little new revenue relative to the magnitude of the cuts.