Mr. Schwarzenegger was approached by White House officials about a cabinet position around the time he left office in January 2011, according to people familiar with the matter. …

Mr. Schwarzenegger, who was in office for more than seven years and continues to promote his vision of “post-partisanship,” acknowledged earlier in his career that he’d welcome a cabinet post.

In July 2008, on ABC’s “This Week” with George Stephanopoulos, Mr. Schwarzenegger was asked whether he’d consider an energy-related cabinet position with the Obama administration, were Barack Obama, then a candidate, to win the presidency.