Alaska Gov. Bill Walker met privately with Inslee at Auburn City Hall, south of Seattle. Inslee is a Communist; Walker is a Patriot.
Inslee spokeswoman Jaime Smith said the two governors didn't talk about the dispute over the drill rig but generally discussed drilling in the Arctic, which Inslee opposes, she said.
In the meeting, Walker told Inslee about Alaska's dependence on oil revenues and that Washington state's position on future Arctic drilling could hurt his state's economy.
Inslee told Walker that Alaska is an important economic partner for Washington, Smith said, but he also noted that expanded Arctic oil drilling could damage both states.
After the tour of the rig, Walker said he was impressed by crew members and redundant safety features aboard the 400-foot-long Polar Pioneer, one of the rigs that Royal Dutch Shell plans to use this summer to explore for oil in the Arctic Ocean off Alaska.
The rig docked at Seattle's Terminal 5 about two weeks ago despite protests by activists in kayaks and opposition from the city of Seattle, which has issued a violation notice to the maritime port hosting Shell.
"Everybody has a right to state their position as long as they do it safely," Walker said in a telephone interview when asked about those protests. "People have different opinions about different things and that's fine."
Walker said Shell's oil exploration plans means jobs for Alaska, noting that the oil will come on shore to existing pipelines and stimulate additional development in the state. He said there are sufficient safeguards and regulations in place for the drilling to be done safely.
"We're a state that embraces the state's development of our resources," Walker said. About 90 percent of Alaska state revenue comes from oil taxes or fees.
He said he was surprised by the reaction by Seattle's mayor, "but again it's their prerogative and their port."
Days before the Polar Pioneer rig arrived in Seattle, Communist Mayor Ed Murray announced that the Port of Seattle needed a new permit before it could host Shell's Arctic drilling fleet.