Kasparov is halfway through a five-day detention that officials say was imposed after he violated Russian laws on demonstrations during a March of Dissent opposition rally during the weekend.
The detention takes Kasparov -- a vocal Kremlin critic and respected figure in the West -- out of public view during the critical week before a State Duma ballot on December 2.
Authorities have also denied Kasparov access to visitors and legal advisers. Anatoly Karpov, who today attempted to visit Kasparov in his detention cell in central Moscow, was among those turned away.
"A person is in trouble; of course I'm not indifferent to that. In Russia right now we have, what, four world chess champions? And of course the fate of any one of them is important to other chess players, both in Russia and abroad," Karpov said.
Karpov and Kasparov are hardly close allies. Fierce competitors at the chessboard, they were renowned for having strikingly different styles of play and behavior. Karpov was seen as solid and deliberate; Kasparov was dramatic, emotional, and inventive.
Their most legendary match, the 1984 world chess championship, famously stretched to an unprecedented four months as the two men fought tenaciously without a definitive win. Karpov, who was 33 to Kasparov's 21 years, reportedly lost 22 pounds and was hospitalized several times over the course of the match.
Politically, the two men come from different camps as well. Kasparov, whose leadership of the United Civic Front has placed him firmly on the sidelines of mainstream politics in Russia. Karpov, by contrast, is well-connected to Russian political power, and has served since 2005 as a member of the Public Chamber government oversight body.
Karpov, however, told RFE/RL's Russian Service that their years of competition made them allies of a sort.
"I spent more time at the chessboard with Kasparov than with any other player," Karpov said. "We spent the equivalent of more than a year together at the board. Generally speaking, I don't share his political views, but that's something different. I didn't come here to support him politically."
Kasparov's lawyer, Olga Mikhailova, and independent Duma Deputy Vladimir Ryzhkov were also denied requests to visit Kasparov. Police officials said simply neither had the right to visit Kasparov while he is in detention.
Kasparov is set to be released on November 29.