WASHINGTON—Congress and President Barack Obama beat the deadline for raising the nation’s debt ceiling by just a few hours on Tuesday, but they hardly ended the clash over the size and reach of government.

The next confrontation promises to be at least as contentious as the one they just finished.

Congressional leaders have two weeks to name members of a special 12-member legislative panel that’s assigned under the debt-limit law Obama signed on Tuesday to find ways to cut the government’s budget deficit by as much as $1.5 trillion by November 23. That number can be reached by reductions in spending and increases in revenues, and the brawl over how to do that already has begun.

“We’ve had too much talk [from] Republican leaders in the Senate saying there will be no revenue. That’s not going to happen,” said Senate Majority Leader Harry Reid, a Democrat from Nevada. “The only way we can arrive at a fair arrangement for the American people with this joint committee is to have equal sharing.”

Republicans signaled that they have a different view, and it doesn’t include higher taxes.

“The answer to this is not giving the government more money to spend,” said Sen. Orrin Hatch of Utah, the top Republican on the Senate Finance Committee. The Senate ended the latest phase of the budget battle on Tuesday by voting 74-26 to increase the $14.3 trillion debt limit—failing to do so could have triggered a government default—and to cut federal spending by trillions.

The first reductions are aimed at reducing federal deficits by $917 billion over the next decade. The bill, which the House of Representatives had approved on Monday, was sent immediately to Obama, who signed it.

The President called the weeks-long standoff over raising the debt ceiling “a manufactured crisis” that didn’t help a faltering economy.

Democrats are furious that tax revenues weren’t part of the package, and the president pledged to continue to press for tax increases to help balance the budget.

“Everyone is going to have to chip in. It’s only fair,” Obama said in a Rose Garden address, singling out subsidies to oil and gas companies and tax loopholes. “That’s the principle I’ll be fighting for during the next phase of this process.”

How the 12-member panel will decide to trim the deficit by $1.5 trillion over 10 years is anybody’s guess. Without an agreement, which Congress then must approve without any amendments by December 23, the debt-ceiling law would impose cuts automatically on so-called discretionary spending programs, including the defense budget.

Indeed, it’s still unknown even how most of the members of the panel will be selected. Senate Republican leader Mitch McConnell plans to interview applicants, but the other legislative leaders—Reid, House Speaker John Boehner, R-Ohio, and House Minority Leader Nancy Pelosi, D-California, haven’t described their selection plans. Each will name three members to the panel.