Google CEO Eric Schmidt last May, introducing Google TV at the company's annual developer conference in San Francisco. (ROBERT GALBRAITH/REUTERS)

Schmidt recently secured a nod from former Reagan administration Commerce Department counsel Clyde Prestowitz. Prestowitz, whose endorsement could signal bipartisan support, said in a Foreign Policy piece that he would “wholeheartedly urge President Obama to move ahead with this appointment.”

Schmidt is known not only for his business acumen but also for his colorful and controversial comments, particularly when it comes to explaining Google’s privacy policies.

The rumors about Schmidt’s possible nomination are apparently substantial enough to make Schmidt’s critics nervous as well.

Consumer Watchdog, well-known for its opposition of Google — and Schmidt in particular — issued a release Thursday in opposition to a possible appointment. The group also sent a letter to Obama asking him not to nominate Schmidt for the position. “Putting Eric Schmidt in charge of policing online privacy is like appointing Bernie Madoff to direct the Securities Exchange Commission,” the release said.

Others who have been mentioned as possibilities for the nomination include U.S. Trade Representative Ron Kirk, Pfizer CEO Jeffrey Kindler and FCC chairman Julius Genachowski.

Current Commerce Secretary Gary Locke is Obama’s nominee to be the next ambassador to China.

Correction: An earlier version of this post incorrectly reported that Google CEO Eric Schmidt was “splitting” with the company. Schmidt will step down as CEO of Google but will remain the company’s executive chairman.