If they don't, Washington's public school system stands to lose control of about $40 million in federal funds.
His proposed law change would likely extend a waiver from the federal government from requirements of the No Child Left Behind Law.
Two different teacher- and principal-evaluation bills are making their way through the Legislature this week. The House proposal, supported by Inslee, would require statewide standardized tests be used in some teacher evaluations, but wouldn't go into effect until the 2017-18 school year. The new rule would go away if the federal government does not extend the waiver.
An alternative bill, proposed by Senate Republican Sen. Andy Hill, would not delay implementation or consider the waiver.
Inslee said he has tried to assemble a consensus "so that we can go to the federal government and win this waiver."
The governor said his proposal has met resistance, "but there is still plenty of time to get this job done and preserve federal funding and the flexibility to serve struggling students in local school districts."
Inslee said he doesn't see a route to get the waiver if lawmakers do not reach an agreement.
Inslee met with U.S. Education Secretary Arne Duncan in Washington, D.C., last week to plead the state's case concerning the waiver and was told Duncan needed to see legislation before he could agree to an extension.
"I've tried to make this clear to legislators. This is a situation where I think all us have to some degree look for a solution even though it might not have been the one we'd design if we were running federal policy," Inslee said.
Hundreds of teachers were in Olympia on Thursday to tell Washington legislators they oppose both bills that would require the use of state test scores in teacher evaluations.
"These two bills are a Band-Aid approach to flawed federal policy," said Elizabeth Kelley, a teacher from Helen Keller Elementary School in Kirkland.
Kelley and other teachers said they're not against accountability, but they have concerns about standardized tests.
Eric Lucke, a science teacher at Finn Hill Middle School in Kirkland, said his fear is that lower-performing schools will not be able to retain the best teachers under this system. They could choose to go to schools where students tend to do better on these tests, which will translate into better evaluations for them, he said.
"It's like telling a dentist, 'We're going to evaluate you on how many cavities your patients have,'" Lucke said. "We need to send a message to federal lawmakers that they need to fix the No Child Left Behind laws."
The waiver system was started by Duncan because he said he couldn't wait any longer for Congress to reconsider the sweeping education law that started the No Child Left Behind system. The law has been waiting for reauthorization for years.
The law says every child in American public schools is required to be reading and doing math at grade level by 2014. The waiver gives states an exception to that rule.
Kelley said she wasn't concerned about losing the waiver.
"The amount doesn't go away. But the district level control would go away, which is a concern. But it's such a low amount of the money spent overall on education."
When asked about the concerns raised by teachers, Inslee said that his goal was to get a bipartisan consensus on the waiver issue, "not argue philosophical arguments about testing, but rather to try to fashion a solution to this immediate problem."
The bills are Senate Bill 5880 and House Bill 2800.
Donna Blankinship reported from Seattle. Associated Press writer Lisa Baumann in Olympia contributed to this report.