According to reports in the national press, protests have been taking place in the capital and across the country since 27 June when a 17-year-old was reportedly killed by a police officer during a traffic stop.
The Guardian, quoting French officials, said a total of 667 people have been arrested across France as of Friday morning (30 June).
The FCDO warned many of the protests have turned violent, with shops, public buildings and parked cars being targeted. It advised some local authorities may impose curfews.
"The protests may lead to disruptions to road travel or targetting of parked cars in areas where protests take place," the FCDO said.
"You should monitor the media, avoid protests, check the latest advice with operators when travelling and follow the advice of the authorities."
French authorities have also banned a rally by groups opposed to the government in Iran planned for 1 July in Paris, due to concerns about security risks; connected events are reported to still be going ahead, however, including in the suburb of Auvers-sur-Oise.
A demonstration by the same group in Paris in June 2018 was the target of an attempted bomb attack.
"British nationals should reconsider any plans to attend such meetings, and if you do, be aware of your surroundings at all times, and move away quickly from disturbances," the FCDO advised.