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Google Chief Executive Eric Schmidt said he will discuss with Apple how his role on its board might change after Google’s move to launch a new operating system.

Because Google’s new Chrome OS would compete with Apple’s own computer software, Schmidt said he would talk to the Mac computer maker about whether he should recuse himself from Apple’s board.

“I’ll talk to the Apple people. At the moment, there’s no issue,” Schmidt told reporters at a Sun Valley media and technology conference organized by boutique investment bank Allen & Co.

While Google and Apple compete directly or indirectly in a number of areas — the most obvious being the smartphone market — Google’s announcement this week that it will launch an operating system raised more questions about the relationship between their boards…

Schmidt began recusing himself from Apple board meetings where the iPhone was discussed after Google launched its own Android mobile phone operating system.

Google’s Chrome OS — expected to debut in netbooks in the second half of 2010 — would compete directly with Apple’s OS X platform.

Schmidt said Chrome, which is based on open-source technology, worked with Apple’s Safari browser.

“There’s a very large collaboration with respect to Chrome and Safari,” he said.

RTFA if you absolutely, positively have to hear about Steve Jobs health.