Mr. Schmidt campaigned for then-Senator Barack Obama during his presidential campaign and advised the transition team and the administration on various matters. He was recently appointed to President Obama’s advisory council on science and technology.

Christine A. Varney, who was recently confirmed as the head of the antitrust division of the Justice Department, last year singled out Google as a probable source of future antitrust concerns because of its near monopoly on Internet search and advertising.

Some antitrust experts said they did not expect Google’s ties to the administration to play a role in antitrust issues.

“I expect the administration to be aggressive, generally, on antitrust enforcement,” said Sanford Litvack, a partner at Hogan & Hartson. Last year, while working for the Justice Department, Mr. Litvack built a case to block a prominent advertising partnership between Google and Yahoo. “I don’t expect Google to either be singled out or to receive a free pass because of Schmidt’s relationship with the administration,” he said.

Antitrust experts say that investigations of interlocking directorates rarely lead to major confrontations between companies and the government. Executives typically choose to resign from the board of a competitor if it poses a problem rather than face a lengthy investigation or a bruising legal fight.

Like many companies in the technology industry, Google and Apple are both allies and competitors. Google, for instance, worked with Apple to design early versions of some its services, like Gmail and Google Maps, for Apple’s iPhone.

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But the areas in which the companies are bumping up against each other as rivals have been increasing.

Mobile phones, in particular, loom large in the future of both Google and Apple. Much of Apple’s fortunes these days are tied to the success of the iPhone. Google, for its part, has said repeatedly that one of its biggest strategic opportunities is to expand its online advertising empire into mobile phones.

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While Google benefits from the success of the iPhone, which drives more traffic to its mobile services than any other device, it also produces the Android operating system for mobile phones that compete with the iPhone. The system currently powers the T-Mobile G1, a phone that some analysts say is the most capable of a number of rivals.

Other phone makers are planning to roll out devices powered by Android later this year. And the Android operating system is being built into lightweight portable computers known as netbooks, which may compete with some Apple laptops.

Google and Apple compete in a variety of other areas. Apple makes the Safari Web browser while Google makes the competing Chrome. Apple’s iTunes and Google’s YouTube are increasingly competing as venues for distribution of music and videos. And the two companies have photo-editing services.

It is not clear whether regulators have singled out any of these areas of competition as particularly troubling. Under the Clayton Act, interlocking directorates are not considered a problem if the revenue from products in which the companies compete is less than 2 percent of either company’s sales.

“Government actions under Section 8 are rare, but they are brought under circumstances when the presence of a common director on competing boards is likely to be anticompetitive,” said Andrew I. Gavil, an antitrust expert and a professor at the Howard University School of Law.

Both Google and Apple share a rival in Microsoft, which competes with the two companies in some areas. But Professor Gavil said regulators were not likely to see that as a problem, even if the two Silicon Valley companies were discussing ways to compete more effectively with Microsoft.

Mr. Schmidt joined Apple’s board in 2006, about five months before it unveiled the iPhone. Google announced its plans for Android, its mobile phone operating system, nearly a year later. Since then, analysts have speculated that Mr. Schmidt’s position on Apple’s board could become untenable. Google has said he recuses himself when Apple’s board discusses mobile phones.