Mr. Burr, along with other senior Republicans like Senator John Cornyn of Texas, the No. 2 Republican in the Senate, has called on Mr. Shinseki to resign, a demand echoed by the American Legion but not by other veterans groups or by Speaker John A. Boehner of Ohio.

But beneath the fight over Mr. Shinseki’s future is a potentially larger war over the Veterans Affairs health care system itself. Over the weekend, the Obama administration said it would allow more veterans to get access to care through private health facilities to ease the backlog at the department’s 152 medical centers and 900 community care facilities.

Republicans in the House and Senate are pushing further. Representative Jeff Miller of Florida, the chairman of the House Veterans Affairs Committee, will introduce legislation this week to give any veteran unable to obtain a V.A. appointment within 30 days the option to go outside the system at the department’s expense.

“For the full basket of reforms to take place at the V.A., there has to be some level of competition,” Mr. Burr said Monday in an interview. “The problems seem to be so systemic, and in some cases, so cultural, that unless there’s a model that gives veterans some ability to decide where to go, I don’t think we will have the means to meet the needs of current veterans, let alone future veterans.”

Veterans groups maintain that the department’s services for spinal cord injury, care for the blind, amputee care and other trauma cannot be matched by private facilities. Over the past two to three years, the department has had a net gain of 1.5 million patients, 200,000 of them with post-traumatic stress disorder and traumatic brain injuries, according to Mr. Sanders.

Relations between the organizations and Senate Republicans have been strained since February, when Republicans blocked a vote on broad veterans legislation, written by Mr. Sanders, that would have bolstered health and dental care, authorized 27 new clinics and medical facilities, added to veterans education programs, and dealt with veterans who suffered sexual trauma while in the military.

Republicans complained that Democratic leaders were not allowing them to amend the bill. But at heart is a budget fight. Republican leaders say the department is plagued by inefficiency and poor management, not insufficient funds. For the past five years, Congress has largely given President Obama what he has requested, and the department has had hundreds of millions of unspent dollars at the end of each year to roll into the next.