Naama Issachar, 26, was arrested on April 9 while in transit in a Moscow airport, en route from India to Israel, and accused of carrying 9 grams of cannabis, her family said. Russian authorities charged her with drug smuggling.
Her family says she is being punished disproportionately to pressure Israel into freeing a Russian held in Israel on a U.S. extradition request.
Netanyahu's request comes after President Reuven Rivlin made a similar appeal to Putin last week.
"Naama made a grave mistake and has admitted her crime, but in the case of a young woman with no criminal record, the severe sentence handed down will have a deeply destructive impact on her life," Rivlin wrote in a letter to his Russian counterpart.
"Because of the particular and individual circumstances of Naama Issachar’s case, I am appealing to your mercy and compassion with a request for your personal intervention to grant her an extraordinary pardon,” Rivlin wrote.
Issachar’s lawyers on Monday officially appealed the sentence.
Naama's lawyer, Alexander Teitz, visited her Tuesday afternoon and said that the young woman "was overwhelmed by the great support she received from Israel and Jews around the world."
The lawyer said his client "had a lot of questions about yesterday's appeal and the ongoing court process. I told her about the possibility of extraditing the Russian hacker to the United States, she received the news with great courage and expressed hope that Israel will ultimately be able to release her from prison in Russia."
Teitz warned Tuesday morning that the eventual extradition of Russian hacker Alexei Burkov would complicate his client's legal situation.
"I want to believe that (the extradition) will not happen, it would be terrible," said Naama's mother, Yaffa Issachar.
"It doesn't make sense, they won't hand him over to America when my daughter is a bargaining chip," she said.