Former Russia international and Chelsea midfielder was criticised in 2015 for comments made to the BBC, when he said: "There's no racism in Russia."
President says he has "full confidence in Russian authorities -- they are taking this matter very, very seriously."
Speaking during a visit to 2022 hosts Qatar, he said the move would be "in line with our sustainability and legacy policies."
Front-page report in L'Equipe said Anelka had allegedly aimed a crude insult at the then national team coach Raymond Domenech.
Russian hooligans have warned England fans that they are "100 percent guaranteed" to be targeted at the 2018 World Cup.
The 68,000-capacity stadium in St. Petersburg hosted 10,000 locals for a free extreme sports show designed to test some of the arena's features.