Washington already has the highest state minimum wage in the country at $9.32 an hour, and it rises steadily thanks to a voter-approved initiative that ties it to inflation.
In his State of the State address Tuesday, Inslee suggested the minimum wage should be somewhere between $10.82 and $11.82 cents an hour.
"There are tens of thousands of jobs that people depend on that don't provide a living wage in our state," the Democratic governor said. "An increase in minimum wage means more money being spent in our economy."
The minimum wage issue has been prominent in Washington state politics recently.
In November, voters in the airport city of SeaTac narrowly approved a measure granting a local $15 an hour minimum wage for many workers. The measure applied to workers at the airport and related industries, like hotels and rental car companies.
But a King County Superior Court judge ruled that the law applied to about 1,600 hotel and parking lot workers in SeaTac, but not to employees and contractors working within Seattle-Tacoma International Airport, which is operated by the Port of Seattle.
Seattle officials have been exploring the possibility of raising the minimum wage there to as high as $15 an hour.
Murray said Inslee's call for a statewide minimum wage increase was a boost for his efforts toward a $15 wage in Seattle. He noted that Seattle salaries tend to be higher than other areas in the state, so he said Inslee's proposed statewide range made sense.
"It's time to have a statewide discussion," Murray said.
Republicans in the House and Senate were cool to the idea, saying it would place strains on the agriculture industry, small businesses and those in counties that border Idaho, which has a minimum wage at just $7.25 - a difference of perhaps $4 an hour under Inslee's plan.
"We need to make sure we're competitive," said House Minority Leader Dan Kristiansen, R-Snohomish.
Other issues Inslee mentioned in his address included:
Inslee wants lawmakers to add another $200 million to the state's education system this year, and he also promised a cost-of-living raise for teachers.
Inslee said the Legislature needs to do more to respond to state Supreme Court demands that the system be more fully funded. Lawmakers did add an extra $1 billion to public schools last year, but Inslee says it is time to do more.
Inslee recently proposed a spending plan that made minimal changes. He says he's had to rethink that approach now that the high court says it wants to see more progress on education funding.
The ruling "has forced us all to look anew at funding our education system this year," he said.
In 2012, justices ruled that the state is not meeting its constitutional obligation concerning education funding. And last week, the high court ruled that lawmakers aren't making improvements quickly enough and have until April 30 to submit a complete plan to detail how the state will fully pay for K-12 education. Lawmakers estimate they need to find a total of between $3.5 billion and $4.5 billion more over the coming years to comply.
"I never envisioned my state, a state that educated so many of us here today, as a place that would need a Supreme Court order to tell it to adequately fund our children's education," Inslee said in his speech. "We need to stop downplaying the significance of this court action. Education is the one paramount duty inscribed in our constitution."
House Majority Leader Pat Sullivan, D-Covington, called that order "a game changer."
"It elevated the need to put more funding into K-12 education this year," he said.
But Senate Majority Leader Rodney Tom, a Democrat who leads a predominantly-Republican majority in the Senate, said it was a matter of simply prioritizing education ahead of other areas of spending.
"If you just keep that limiter, we can more than fund this without new revenue," Tom said.
Inslee said this is the year to pass a transportation spending package. Last year the state Senate declined to take a vote on a package approved in the state House. Inslee said if the Senate can pass a package of its own, he's confident lawmakers can find agreement on the spending plan.
Leading lawmakers have said a spending package that increases the state gas tax may be difficult to pass this year because of ongoing problems at the Department of Transportation. Officials are dealing with cost overruns on a 520 bridge project and a tunneling machine that is stuck in Seattle.
Inslee says he is also frustrated with the problems at DOT, but he says the state can't let issues on megaprojects stop it from moving forward.