In a message on its website, the Roskomnadzor agency warned that "time is running out" for Telegram to comply with the law.
Otherwise "Telegram must be blocked in Russia," it said.
The free instant messaging app lets people exchange messages, photos and videos in groups of up to 5,000 people. It has attracted about 100 million users since its launch in 2013.
In an unusual move, the agency's chief, Alexander Zharov, addressed Telegram's secretive Russian chief executive, Pavel Durov.
The watchdog said it had been sending messages to the company "practically every day" asking it to send company information for Roskomnadzor's register.
Durov has not responded on his social media accounts.
The 32-year-old had previously created Russia's popular VKontakte social media site, before founding Telegram in the United States.
Durov said in April that the app had "consistently defended our users' privacy" and "never made any deals with governments."
But the service has drawn the ire of critics who say it can let criminals and terrorists communicate without fear of being tracked by police, pointing in particular to its use by Islamic State jihadists.
Since January 1, Internet companies are required to store all users' personal data at data centres in Russia and provide it to the authorities on demand.
And draft legislation that has already secured initial backing in parliament would make it illegal for messaging services to have anonymous users.
A bill set for its first reading Friday would also ban the use of virtual private networks (VPNs) and proxy servers to bypass Russian internet controls.
But Russia's internet ombudsman Dmitry Marinichev said the VPN bill was "going against common sense", speaking to RBK independent news site this month.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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