A grass-roots campaign over Congress' August recess has increased pressure on the leaders to attach the health care provision, but a Boehner spokesman said no decision has been made.
Congressional Democrats and the White House are eager to reverse the cuts, and many defense hawk Republicans would like to as well. But there have been no fruitful negotiations between the White House and House GOP leaders.
Negotiations between White House officials and a small group of Senate Republicans collapsed last month over familiar disagreements over tax increases and cuts to popular federal benefit programs.
Without a deal, those automatic spending cuts could become entrenched through all of next year and possibly into the next several years.
A 2011 agreement called for a total budget of $1.058 trillion next year to operate federal agencies. The automatic spending cuts triggered by failing to follow up with further deficit cuts by curbing benefit increases, raising taxes or both would pare that figure by $91 billion, to $967 billion for the 2014 budget year.
A comparable spending figure for the soon-to-be-completed 2013 budget year is about $988 billion. The additional cuts looming next year come almost entirely from defense.
House Armed Services Committee Chairman Howard "Buck" McKeon, R-Calif., says many Republicans on his committee want to reverse the defense cuts as a condition for voting for the resolution authorizing military strikes on Syria.
Avoiding a shutdown is just one step.
The administration says the government's ability to fully pay all its bills, including interest payments to bondholders and Social Security benefits, will run out some time in October unless Congress raises the $16.7 trillion cap on its borrowing authority.
That legislation could be even more vexing because Boehner and tea party Republicans see it as leverage to force further spending cuts or other GOP priorities into law.
Obama agreed in 2011 to Boehner's demand that spending cuts equal the size of the debt limit increase, but the president says he won't do it again. Republican leaders say such a "clean" debt limit increase Obama wants is a nonstarter.
An immigration overhaul could get lost in the shuffle.
The Senate in June passed a broad bill that would allow millions of immigrants now in the country illegally to stay, work and eventually acquire citizenship. House Republicans reject what they call a special path to citizenship in the Senate bill and favor a piecemeal approach that begins with better securing U.S. borders before excusing most people who entered the country illegally or overstayed their visas from being deported.
Revelations this summer about the National Security Agency's spying prompted demands from some in Congress to rein in the programs; a series of hearings is scheduled. Leaders of the House and Senate intelligence committees face pressure from lawmakers to make changes to NSA surveillance in the longstanding authorization bills.
Lawmakers also are pushing for major changes in how the military handles cases of sexual assault in their ranks. A significant number of lawmakers want to remove commanders from the process of deciding whether serious crimes, including sexual misconduct cases, go to trial.
The Pentagon and some senior lawmakers reject that idea. A showdown is expected later this year when the Senate debates a defense policy bill.
Congress also will have to finish a farm bill before the end of the year if lawmakers want to avert the threat of milk prices doubling for consumers. Most of current farm law expires at the end of September, but its effects won't be felt until the end of the year when dairy supports expire. Without the supports, milk prices are expected to rise.
Associated Press Writer Mary Clare Jalonick contributed to this report.
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