YouTube and Instagram may be blocked in Russian Federation as early as Wednesday unless they comply with the order and remove the material, Vedomosti newspaper reported.
The videos in question are from the social media accounts of blogger Anastasia Vashukevich, who posted the videos of Deripaska and Prikhodko on the yacht. She claimed in a book she wrote that she was hired as an escort on Deripaska's boat, but her allegations weren't taken seriously until Navalny's expose last week, according to The New York Times.
The Guardian writes that Navalny claims Deripaska "was briefing Prikhodko on information on the 2016 US presidential campaign that he had received from Paul Manafort, the former Trump campaign manager", although he offers no proof of this.
YouTube and Instagram did not immediately respond to NPR's requests for comment.
Navalny's website, the webpage dedicated to his investigation concerning Deripaska and Deputy Prime Minister Serge Prikhodko, as well as the relevant YouTube video, were earlier included in the list of information banned from circulation.
Russia has permanently blocked the popular online connection site LinkedIn after the firm refused to move servers to Russian territory and open up the information to Russian officials.
A February 14 deadline that Navalny said was set by Roskomnadzor passed without incident, but reports that navalny.com was inaccessible for many in Russian Federation began flowing in on February 15.
Roskomnadzor said it had informed the sites that they must delete the images within three days.
Navalny said: "It wasn't me who took bribes from an oligarch in the form of aeroplane flights and expensive cruises, but the singular, lightning-fast attack of the government is directed against me and my foundation, who exposed the corruption".
Navalny's report caused a potentially embarrassing splash ahead of a March 18 election in which Putin, who has been president or prime minister since 1999, appears certain to win a new six-year term but is said to be hoping for a convincing victory that would strengthen his mandate and further cement his power.
Roskomnadzor blacklisted the posts after Deripaska filed a lawsuit, prompting a court injunction demanding their removal, according to The Moscow Times. 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.
"The tactic may signal a more aggressive approach by the Russian government in its bid to rein in social media and video-sharing websites popular in the country".