WASHINGTON — House and Senate negotiators opened compromise talks Tuesday on legislation to expand health care for veterans, hoping for a quick response to a scandal that has uncovered long wait times, false record-keeping and accusations of criminal activity at the Department of Veterans Affairs.

At their core, the bills passed by the House and Senate would allow millions of former members of the armed forces to seek health care outside the government’s system if they were unable to get a timely appointment inside it.

“The simple truth of the matter is that the VA needs more doctors, more nurses, more mental health providers, and, in certain parts of the country, more space for a growing patient population,” said Sen. Bernie Sanders, the Vermont Independent who chairs the Senate veterans committee.

Rep. Jeff Miller, R-Fla., and chairman of the House panel, noted the deaths reported among veterans awaiting care in Phoenix and said that for them, “our work begins too late. For veterans still waiting, our work begins” none too soon.

The lawmakers met one day after the latest indication of a federal agency in deep trouble. In a letter to President Barack Obama, the U.S. Office of Special Counsel said the agency shows a “troubling pattern of deficient patient care” and also pays little attention to reports from its own whistleblowers.

The scandal — which cost former secretary Eric Shinseki his job — erupted this spring with reports — as yet unproven — that a delay in care at the agency’s facility in Phoenix contributed to the deaths of about 35 veterans.

It quickly mushroomed when a VA report concluded about 1,700 veterans in need of care were “at risk of being lost or forgotten” at the center after being kept off an official, electronic waiting list. Then came reports of problems at numerous facilities around the country.