Twitter had “expressed its readiness and interest in building a constructive dialogue with Roskomnadzor”, the watchdog added.
It “appreciates the efforts of Twitter to comply with the requirements of Russian legislation”, it said.
But for all the restrictions to be removed it would have to remove “all identified prohibited materials”. It would keep slowing Twitter on mobile devices, it added.
Roskomnadzor did not say if it would stop slowing Twitter’s services if the company deleted the rest of the content and did not give a deadline for doing so.
Facebook, YouTube warned
Roskomnadzor also said that it had identified cases of illegal content “on other internet sites, including Facebook and YouTube.”
“In the event that these platforms do not take appropriate measures, similar sanctions may be applied to them,” the watchdog said.
The measures against Twitter and other social media have raised concerns among Kremlin critics, who fear the clampdown is aimed at silencing opposition voices.
Last month a Moscow court imposed on Twitter three fines totalling 8.9 million rubles ($120,000) for failing to delete posts calling for minors to join unsanctioned protests in support of jailed Kremlin critic Alexei Navalny.
A Moscow court that month also hit TikTok with a fine of more than $30,000 for the same violation.
In January, the authorities accused foreign social media platforms of interfering in Russia’s domestic affairs by not deleting calls to rallies in support of Navalny.
President Vladimir Putin that month complained about the growing influence of large technology companies, which he said were competing with states.