The transport strike has been going on for 38 days. The protests concern the pension reform that has been proposed.
Prime Minister Édouard Philippe now declares his willingness to withdraw the most disputed proposal in the reform. The proposal would mean working at age 64 to receive full pension.
French President Emmanuel Macron calls the offer "a constructive and responsible compromise".
The Prime Minister's play receives a mixed reception from the unions. CFDT and UNSA welcome it. But the more uncompromising allies CGT and FO are thumbing down.
CGT says the offer is a "smoking curtain to get some unions under" on the reform.
On Saturday, new demonstrations were held in Paris against the proposed pension reform. Some of the protesters were masked. They shattered shop windows and threw things at the police, who responded with tear gas.
Demonstrations were also organized in Marseille, Toulouse, Lyon and Nantes, among others.
According to the Home Office, nearly 150,000 people participated in the demonstrations across France. But according to CGT, there are 500,000 people who protested.