Russia's internet censor blacklisted material on both services after a court ruled that it violated billionaire Oleg Deripaska's privacy rights.
However, Alexei Navalny has refused to remove the videos and photos, which he claims are evidence of corruption.
A Wednesday deadline has been set.
If neither Mr Navalny nor the US tech firms involved delete or otherwise block local access to the imagery by the end of the day, then Russia's ISPs will be required to take action themselves.
A group representing the industry has indicated that this could result in all local access to the social networks being curtailed since ISPs lack the facility to censor specific posts.
"It's impossible for internet providers to block certain pages on Instagram and YouTube," a spokeswoman for the Russian Association for Electronic Communications told the BBC.
Mr Navalny is Russia's most prominent opposition leader. But he has been barred from standing against President Putin in next month's election because of a corruption conviction, which he says is politically motivated.