Conducted last week -- while ballots were going out to state households -- the survey showed that Democrat Jay Inslee and Republican Rob McKenna each had the support of 46 percent of likely voters.
Inslee was faring better among women, while McKenna was getting the support of 19 percent of people who voted for President Barack Obama in 2008.
The poll of 500 likely voters was conducted by consulting firm Strategies 360. It has a margin of error of plus or minus 4.4 percentage points.
Respondents indicated they were supporting the state's leading ballot measures, with a gay marriage initiative seemingly heading toward passage with 55 percent in favor and 38 saying they would reject it.
However, only 43 percent of respondents said they "strongly" approved of legalizing gay marriage.
Kevin Ingham, vice president of polling and research at Strategies 360, said ballot measures typically lose some of their "yes" votes as Election Day approaches, and he suspects that people who say they are undecided on the gay marriage issue are likely to oppose it.
An initiative to legalize marijuana was also leaning toward passage, with a 54-38 percent advantage. But some results indicated people appeared uncertain on the measure, with 34 percent saying they strongly support the proposal while 20 percent said they only "somewhat" support it -- something that Ingham said may worry the initiative's backers.
A measure to implement charter schools in Washington state was leading 51-34 percent, according to the poll. About 38 percent of respondents said they were strongly in favor of the plan. Ingham said it will take a lot of work for the campaign to keep the "yes" votes above 50 percent.
Pollsters also explored how the state should respond to a state Supreme Court ruling that determined the state isn't properly funding education. Lawmakers are trying to find money to respond to that ruling, and some have proposed new taxes.
Respondents indicated strong opposition to broad taxes, with about two-thirds opposing any increases in sales, property or business taxes. The likely voters were most receptive to a proposal to implement an income tax on people making over $200,000 -- approving that idea by a 54-41 percent margin. However, voters resoundingly rejected a similar plan just two years ago.
Inslee and McKenna have said they would oppose new taxes and believe they can find other ways to fund education.
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