Naama Issachar trod a path familiar to many Israelis after completing her army service—she spent a few months traveling around India and planned to return to Tel Aviv in April, flying through Moscow because the airfare was cheaper.

That’s when Russian authorities found a small amount of marijuana—9 grams, or slightly less than one-third of an ounce—in her suitcase. Ms. Issachar, a 26-year-old Israeli-American, never boarded the plane back to Israel and has been detained in Russia ever since.

In August, Ms. Issachar’s family members began to receive strange messages from a man who said he was friends with an alleged Russian hacker, Alexei Burkov, who has been detained in Israel since 2015 at the request of the U.S., where he is said to be wanted on cybercrime charges related to alleged credit-card fraud.

It eventually became clear that the fates of Ms. Issachar and Mr. Burkov had become intertwined. Russia was seeking to engage Israel in a prisoner exchange to retrieve one of its own, according to people familiar with the matter.

For months, Moscow has pursued what current and former U.S. law-enforcement and diplomatic officials describe as part of a stepped-up and evolving campaign to prevent Russians arrested on criminal hacking charges from being extradited to the U.S.