The 24-year-old was traded to Columbus on July 23 along with Brandon Dubinsky, Tim Erixon and a 2013 first-round draft pick by the New York Rangers in exchange for Rick Nash, Steve Delisle and a conditional third-round pick.
In 79 games with the Rangers last season, Anisimov registered 16 goals and 20 assists for 36 points.
Meanwhile, Blue Jackets goaltender Sergei Bobrovsky hasn't decided when he'll head to North America. He will, however, return at some point this week.
Bobrovsky, 24, has been playing with SKA St. Petersburg of the KHL during the lockout.
The native of Novokuznetsk, Russia, is expected to return posted an 18-3-2 record with SKA this season, along with a 1.94 goals-against-average and a .932 save percentage.