WASHINGTON — U.S. military veterans would have easier access to medical care under a rare bipartisan agreement passed Wednesday by the House and set for final Senate passage by the end of the week.

The measure, approved 420-5, would cost about $17 billion. It would create 27 new Department of Veterans Affairs medical facilities and expand care for veterans at non-VA hospitals and clinics. It also would allow the VA secretary to fire senior executives.

The show of bipartisanship in Congress is so unusual that House Speaker John Boehner, R-Ohio, described House Veterans Affairs Committee chairman Jeff Miller, R-Fla., who negotiated the deal, as a hero.

Mr. Boehner said Republicans supported the measure, which will add to the U.S. budget deficit, to ensure that veterans will have access to private care. The bill would be financed with about $12 billion in emergency funds and $5 billion in offsets elsewhere in the VA budget, said Mr. Miller and Senate Veterans Affairs Committee chairman Bernie Sanders, I-Vt., who caucuses with Senate Democrats.

The five Republicans voting no were Reps. Rick Crawford of Arkansas, Walter Jones of North Carolina, Jack Kingston of Georgia, Mark Sanford of South Carolina and Steve Stockman of Texas.

Increased payments in the deal could triple health care providers’ revenue from the VA in the next year, according to a Bloomberg Intelligence analysis. In 2013, HCA Holdings Inc. and Universal Health Services Inc. hospitals had the most admissions for veterans’ care among publicly traded hospitals in California, Texas and Florida, states with the most veterans.

Senate Majority Leader Harry Reid, D-Nev., has said he is optimistic that his chamber can pass legislation before lawmakers leave Washington for a five-week break at week’s end. The Obama administration also welcomed the deal, White House press secretary Josh Earnest said.

The VA deal seeks to contain costs by limiting eligibility for non-VA care to veterans already enrolled in VA health care programs, those who live far from veterans’ medical facilities and others who cannot quickly get an appointment at a VA clinic or hospital.

Former VA Secretary Eric Shinseki resigned in May after an inspector general report showed widespread mismanagement, such as keeping phony lists to hide the long waits that veterans face for medical appointments. At least 35 veterans died while awaiting care in Phoenix, according to acting VA Secretary Sloan Gibson. The Senate on Tuesday unanimously confirmed former Procter & Gamble Co. chief executive officer Bob McDonald to lead the VA.

The VA, with a $160 billion budget, runs the nation’s largest integrated health care system. An internal audit in June showed that more than 120,000 veterans hadn’t received a medical appointment or were waiting more than 90 days for care. That number was reduced to about 42,400 by July 1, VA data show.

“There’s thousands and thousands of veterans who have not been able to access our local hospital to get health care that they have been promised,” House Republican Conference chairwoman Cathy McMorris Rodgers, who is from eastern Washington state, said Tuesday. “The way that we show our gratitude to our veterans is by making sure that they are taken care of upon their return.”