Recent court documents provide the clearest indication yet that special counsel Robert Mueller is zeroing in on the ties between WikiLeaks and Roger Stone, a former Trump campaign adviser whose relationship with the president dates back nearly four decades.

According to the documents and more than a half-dozen people who have testified in the probe, Mr. Mueller’s office in recent months has focused on Mr. Stone’s role as a potential conduit between the Trump presidential campaign and WikiLeaks, which during the 2016 election...