WASHINGTON — Veterans Affairs Secretary Eric Shinseki called in leaders of veterans groups Thursday and pledged to keep VA hospitals open nights and weekends if necessary to set up speedy appointments for veterans whose long waits for medical care triggered a departmental crisis.

As bipartisan pressure mounted in Congress for his resignation, Shinseki said he would announce as soon as Friday new steps to hold VA employees accountable, including personnel changes and the option for veterans to seek private-sector care, according to participants at the hourlong meeting.

“I got the idea that there might be some heads that will roll,” said Garry Augustine, executive director of the Disabled American Veterans’ Washington office. “I think they realize the urgency.”

Shinseki’s fate could be determined by the preliminary findings of an audit he ordered of the VA health-care system, which he is expected to deliver to President Obama on Friday.

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A report from the VA’s Office of Inspector General on Wednesday found systemic problems throughout the hospital network in scheduling veterans for medical care, including manipulating records to hide long waits for appointments.

The report also disclosed that the investigation had expanded to 42 sites, up from the 26 previously stated.

In Texas, meanwhile, Republican Sen. John Cornyn said he sent a letter to Obama calling for the FBI to lead the investigation into the accusations.

At the White House, spokesman Jay Carney stopped short of declaring Obama’s full confidence in Shinseki, a retired four-star Army general, saying the president has “made clear that he believes there ought to be accountability once we establish all the facts.”

The firestorm surrounding Shinseki heated up, as at least 100 members of Congress, including almost a dozen Senate Democrats, called for Shinseki to step down.

But House Speaker John Boehner, R-Ohio, and Minority Leader Nancy Pelosi, D-Calif., both said replacing Shinseki would not solve the core issue of unacceptable wait times and secret waiting lists at Department of Veterans Affairs health centers.

Boehner said: “The question I ask myself is, is him resigning going to get us to the bottom of the problem? Is it going to help us find out what’s really going on? And the answer I keep getting is no.”

Shinseki was getting support from some quarters. Stewart Hickey, national executive of AMVETS, the veterans service organization, pleaded with House Republicans that “cutting off the head of the monster” would not solve the problems veterans are facing.

Derek Bennett, chief of staff of the Iraq and Afghanistan Veterans of America, one of the veterans groups that met with the VA chief Thursday, said Shinseki “did not seem to me to be shaken by the recent drumbeat from both sides of the aisle for his resignation.”

But Bennett said the meeting “did nothing to restore confidence” in the secretary to fix the problems.

Cornyn, who met in Houston with about a dozen local veterans and leaders of veteran groups Thursday, said he wanted the FBI to investigate allegations by Texas VA employees that managers received bonuses after concealing wait times for appointments.

“When you hear the leadership at the VA are receiving bonuses only for cooking the books, to me that speaks of corruption,” Cornyn said.

Sen. Mark Warner, D-Va., said: “The controversy over Gen. Shinseki’s leadership has taken attention away from the real issue: the need for swift, decisive action to reform the VA, change its culture and ensure that we provide quality, timely services for our veterans.”

To that end, House Republicans readied legislation that would send veterans who have waited for care for 30 days or more to private physicians, at the VA’s expense.

Senate Democrats were drafting a far more complex bill. To alleviate the short-term crisis, it would allow veterans to get private care paid for by the VA, and use Defense Department health centers, Indian Health Services and federally licensed community health centers. Longer term, it would pay for 27 new VA facilities and advance a partnership between the VA and the National Health Service, offering medical students scholarships or loan forgiveness in exchange for five years of service at VA facilities, said Sen. Bernard Sanders, I-Vt., chairman of the Senate Veterans Affairs Committee.

“If there’s any silver lining in this horrendous situation, maybe the country will focus on the crisis of primary care throughout our health system,” he said.

Material from The New York Times is included in this report.