WASHINGTON—Special counsel Robert Mueller’s probe is looking more closely into Middle Eastern involvement during Donald Trump’s 2016 presidential campaign, as it is now exploring the role of an Israeli entrepreneur with ties to a Gulf monarchy, according to people familiar with the matter.

Mr. Mueller has been conducting interviews about the work of Joel Zamel, an Australia-born Israeli businessman with experience in social media and intelligence gathering. Mr. Zamel is the founder of several private consulting firms—including...