Germany's justice minister has drafted a law that seeks to impose the fines as part of efforts to police toxic chat.
Heiko Maas said the voluntary efforts of social networks to tackle the problem had not gone far enough.
The proposal requires sites to run 24-hour helplines and to delete flagged content within seven days.
Social media firms such as Twitter and Facebook were getting better at handling illegal content, said Mr Maas, but both had a long way to go.
Mr Maas quoted research which suggested Twitter deletes only 1% of the hate speech it is told about by users, and Facebook, 39%.
"This isn't sufficient yet," said Mr Maas.
Racism and hate speech are believed to have become more prevalent on German social media following the arrival of large number of refugees in Germany.
Any content that was "clearly criminal" would have to be removed within 24 hours under conditions outlined in the draft law. If, after an investigation, content is found to be criminal then that must be removed in seven days. The people who posted the illegal content must also be told about its deletion.
The proposed law would require each network to run fully staffed, round-the-clock reporting systems and to name an individual responsible for handling complaints. That person could face an individually levied fine of five million euros if companies break laws governing what can be published.
Mr Maas said the law could apply to fake news articles if they proved to be slanderous, defamatory or libellous.
Facebook did not comment directly on the proposal but said tests it commissioned showed it removed a higher percentage of illegal content than Mr Maas claimed. The social network said it expected to have 700 people employed in Berlin by the end of 2017 overseeing its efforts to review flagged content.
German digital trade association Bitkom criticised the proposed law. It told the Financial Times that the requirement to remove material within 24 hours on sites that handle more than one billion posts per day was "utterly impossible to implement in operational terms".