The incident comes amid an ongoing Department of Justice investigation into Thompson for alleged antitrust violations and accusations of insider trading.
The suspect has been described as a white male wearing a cream-colored jacket, a black face mask, black and white sneakers, and carrying a grey backpack.
According to Los Angeles Police Chief Jim McDonnell, Kobayashi is believed to have arrived at the San Ysidro border crossing on November 12, where she crossed into Mexico.
During President Biden's time in office, she has developed a reputation for managing who gets access to him and taking assertive measures to protect him from media scrutiny.
John, who was joined by his husband, composed the original score for "The Devil Wears Prada: The Musical," which features Tony nominee Vanessa Williams as Miranda Priestly.