Moderate Republicans from the Northeast have found such cuts impossible to swallow, and when their numbers were combined with ardent conservatives who never support appropriations bills, leaders have found themselves short by dozens of votes. Representative Jim Gerlach, Republican of Pennsylvania, said he could not accept Amtrak cuts, which would have reduced rail financing by one-third, and cuts to Community Development Block Grants below the level set when the program began under President Gerald R. Ford.

For appropriators, “the frustration is, ‘Hey guys, you wrote the budget. We don’t do that. You asked us to produce the bills. We’re taking very difficult votes here. And at the end of the process, you’re not able to move these bills. You have to rethink your strategy,’ ” said Representative Tom Cole, Republican of Oklahoma, who sits on the Appropriations Committee.

The House Appropriations Committee chairman, Representative Harold Rogers of Kentucky, released a remarkable broadside against his own leaders.

“The House has declined to proceed on the implementation of the very budget it adopted just three months ago,” he wrote. “Thus, I believe that the House has made its choice: sequestration — and its unrealistic and ill-conceived discretionary cuts — must be brought to an end.”

The White House-Senate negotiations center on restoring domestic and military spending at least to the limits set by the Budget Control Act, and shifting those savings to entitlement programs and other “mandatory” programs not covered by the appropriations bills. Such a trade-off would be hard for the White House; Mr. Obama and senior advisers have said that he would not support reductions from the health-related entitlement programs, Medicare and Medicaid, unless Republicans agree to raise additional revenues.

Ms. Burwell called the public display of Republican division a potential “inflection point” that could force a change in their unyielding budget stand. And she rejected the conventional wisdom in Washington that even if the White House reached a deal with Senate Republicans, House Republicans would reject it. “You make the reality,” she said.

While Ms. Burwell echoed Mr. Obama in asserting that the White House had not given up on a comprehensive and long-term budget deal after two years of failures, administration officials lately have been consulting with Republicans about more limited accords to reduce deficits while investing now in programs to bolster the middle class.