The request to block the sites was made by the head of communications watchdog Roskomnadzor, Alexander Zharov, at a meeting with Colin Crowell, Twitter's head of global public policy where the two men discussed new Internet regulations.
Following the talks, Zharov was quoted by Russian news agencies Itar-Tass as saying: "It does not matter where the blog has been registered ... I hope that these accounts will be deleted as soon as possible."
The report did not identify the Twitter accounts concerned.
Last month, Twitter blocked access in Russia to an account linked to a far-right Ukrainian nationalist group, days after an official at the regulator threatened to cut off the micro-blogging site completely if it did not comply with new rules that allow the government to ban sites without a court order.
Russian Prime Minister Dmitry Medvedev denied at the time there was any plan to close down Twitter.
A spokesman for Twitter, Nu Wexler, confirmed Monday's meeting had taken place, to discuss the new legislation, but said it had not agreed to block any further accounts in Russia.
One of several new laws on the Internet requires firms to keep servers handling Russian traffic inside the country and store information about users for at least six months.
Russian President Vladimir Putin, who has described the Internet as a "CIA project", signed a law last month requiring blogs with more than 3,000 daily visits to register with Roskomnadzor and adhere to rules governing the mass media.
The Kremlin, which denies allegations of media or Internet censorship, also adopted a law earlier this year giving authorities power to block websites deemed either extremist or a threat to public order without a court ruling.
Among those websites blocked under the new rules were those of Kremlin critics Alexei Navalny and Garry Kasparov because they "contained calls for illegal activity".