At his first official news conference as governor, Inslee said that while he's not proposing anything specific, he wants to give the Legislature "room to discuss this potential."
"I don't want to foreclose the possibility of those being on the table for discussion," he said. "I'm not proposing it right now. I think it's something that people are ultimately going to consider."
Last month, former Gov. Chris Gregoire proposed a budget that would extend taxes on beer and business taxes paid by doctors, lawyers, accountants and others.
Inslee, a Democrat, said he's not breaking his campaign promise eschewing taxes because he doesn't consider the extensions new taxes.
"These do not increase taxes," he said. "They do not raise taxes on people over the existing level that they are paying today."
But House Republicans' point person on the budget, Rep. Gary Alexander of Olympia, said his caucus considers extending taxes the same as a tax increase and oppose any move in that direction.
"We're hoping we can reach a budget solution that doesn't require any tax increases and that the governor will support it when we do," he said.
Inslee was sworn in Wednesday. He takes office facing a projected $900 million deficit for the next two-year budget ending in mid-2015.
That doesn't include money lawmakers will need to spend to improve funding for education as directed by the state Supreme Court earlier this year.
House Majority Leader Pat Sullivan, D-Covington, said that considering the large fiscal problem ahead of lawmakers, Inslee's clarified stance on extending temporary taxes "certainly makes it easier, if that's something we can consider as part of a solution."
"When you're looking at the range of options, that's one that makes some sense," he said.