Russian authorities have given no information on her whereabouts for nearly two weeks, but the source said she had been taken to Female Penal Colony IK-2 in Yavas, about 500 km southeast of Moscow.
The colony is in Mordovia, the same region where another American, Paul Whelan, is serving a 16-year sentence in a different penal settlement after being convicted of espionage charges that he denies.
Asked about Griner's case, a U.S. State Department spokesperson said: "We are aware of reports of her location, and in frequent contact with Ms. Griner's legal team.
"However, the Russian Federation has still failed to provide any official notification for such a move of a U.S. citizen, which we strongly protest. The Embassy has continued to press for more information about her transfer and current location."
In August, Griner was sentenced to nine years in a penal colony on drugs charges following her arrest at a Moscow airport in February with vape cartridges containing cannabis oil in her luggage. She was moved from a detention center near Moscow on Nov. 4.
Russia and the United States have discussed swapping Griner and Whelan, a former U.S. Marine, for a Russian arms dealer jailed in the United States, but no deal has materialized amid heightened tensions over Russia's invasion of Ukraine.