“Frankly, the fall is shaping up to be the most predictable — and, really, avoidable — budget crisis in memory,” said Representative Nita M. Lowey of New York, the ranking Democrat on the House Appropriations Committee.

Two changes have combined to make this budget season particularly contentious. The first is Republican control of Congress. Senator Mitch McConnell, the majority leader, said the appropriations process was a way for Republicans to block new White House efforts and roll back old ones.

The other change is budgetary. After a protracted fight over raising the government’s statutory borrowing limit, Congress and Mr. Obama agreed in 2011 to strict spending limits for domestic and defense programs that would last a decade, absent a larger budget deal that overhauled the federal tax code and entitlement programs like Medicare. In the face of those limits, Representative Paul D. Ryan, Republican of Wisconsin, and Senator Patty Murray, Democrat of Washington, negotiated a two-year budget deal to lift the caps.

With the reprieve ending Oct. 1 and Democrats no longer in control of the Senate, Republicans approved a budget this spring demanding adherence to the 2011 restrictions.

“The Appropriations Committee in the Senate is reporting out bills consistent with the budget that the Senate passed, and we’re going to move forward,” Mr. McConnell said last month.

With Democrats promising to vote against any bill that sticks to those limits, Republican appropriators have had to shift spending bills to the right to win the support of Congress’s most conservative members. Mr. Cole called the 10 bills that have passed the Appropriations Committee “the opening position in a negotiation” that has yet to begin.

But that position has only strengthened the president’s resolve.

Compounding the problem is the fact that the government will be nearing its statutory borrowing limit in late September, when Pope Francis is to be in Washington, heightening tensions around cuts to social welfare programs.