Special counsel Robert Mueller asked questions about the relationship between President Trump and the billionaire Russian-Azerbaijani family who arranged the June 2016 meeting between Trump campaign aides and a Russian lawyer linked to the Kremlin, according to a participant of the meeting.

Rob Goldstone, who worked as a publicist for the Agalarov family and contacted Donald Trump Jr. in June 2016 on their behalf, spent roughly eight hours talking to Mr. Mueller’s team in March. Weeks later, he testified before the special...