(Reuters) - The Senate on Wednesday overwhelmingly passed a bipartisan measure aimed at easing healthcare delays for veterans by giving them more access to private care and allowing the Department of Veterans Affairs to open more clinics and hire more medical staff.

The 93-3 vote in the Democratic-led Senate followed unanimous passage on Tuesday in the Republican-controlled House of Representatives of a similar bill to address a crisis that has embarrassed the Obama administration and prompted Veterans Affairs Secretary Eric Shinseki to quit.

Lawmakers must now iron out differences between the House and Senate versions before voting on a final package that could be signed into law by President Barack Obama.

The Senate measure matches several provisions passed by the House to address a crisis unfolding in the run-up to November's congressional elections.

The senators who authored the bill emphasized the need for urgency. It was crafted by Senate Veterans Affairs Committee Chairman Bernie Sanders, an independent from Vermont, and Republican John McCain of Arizona, a state where 40 veterans are said to have died while waiting months for appointments at VA clinics in Phoenix.

If there is a definition of emergency, I would say that this legislation fits," McCain said.

"We've got to get a good bill on the president's desk next week," Sanders said.

Before passing the bill, senators voted 75-19 to turn aside objections to its cost raised by Republican Jeff Sessions of Alabama. "I feel strongly we've got to do the right thing for our veterans. But I don't think we should create a blank check, an unlimited entitlement program, now," Sessions said.

Sessions and two other Republicans, Ron Johnson of Wisconsin and Bob Corker of Tennessee, voted against the bill's passage.

Provisions passed by both chambers would allow veterans to visit private doctors at VA expense if they are forced to endure long waits for appointments at VA clinics or live more than 40 miles (64 km) away, and would give the VA secretary more power to fire or demote employees for poor performance.

The Senate measure also matches earlier House-passed legislation that authorizes the VA to sign leases for 26 new clinics in 18 states. The bill would require an emergency supplemental appropriation, which Sanders estimated at under $2 billion, mainly for the opening of the 26 clinics. Among differences between the House and Senate versions are that the House proposes a top-to-bottom review of all aspects of VA's health care system, while the Senate calls for a review of appointment scheduling practices and systems. The Senate version offers protections for VA employees not in the House bill.

The VA operates the largest U.S. healthcare system, with 151 hospitals and 827 outpatient clinics serving 8.9 million veterans.

The FBI said on Wednesday it has opened a criminal investigation of an Arizona VA office.