Workers at Exxon Mobil France's Port Jerome and Fos refineries began a four-day strike, the hardline CGT union said.
An Exxon spokeswoman confirmed that the 140,000 barrels-per-day (bpd) Fos-sur-Mer plant, which accounts for about 10% of French refinery output, was blocked but added that the 240,000 bpd Port Jerome refinery was operating normally.
French Prime Minister Edouard Philippe said the country was not at risk of fuel shortages and that police would ensure oil depots were not blockaded.
"People have the right to strike, but they do not have the right to block (refineries)," he said on RTL radio, referring to picketing by workers to obstruct factory gates.
Philippe said he was open to discussing changes to the average retirement age with unions, one of the main sticking points over pension reform plans that have triggered protests.
The environment ministry said all French refineries continued to operate, but five out of seven were temporarily having difficulty distributing their products.
It said it expected no problems with petrol stations, whose supplies are guaranteed by a separate network of 200 depots.
On Tuesday, only three of these depots reported difficulties while others were operating normally, the ministry said, adding that France has stocks corresponding to more than three months of fuel consumption.
A spokesman for French petrol industry lobby UFIP said that nationwide only about 155 out of 11,000 petrol stations - about 1.5% - experienced shortages of some products.
"This is mainly due to panic buying, not to a shortage of fuel," oil expert Jean-Louis Schilansky said.
FNIC-CGT union leader Emmanuel Lépine said on France Info radio that seven out of eight refineries were on strike. The 96-hour strike is set to last until Friday.
Total said that only about 5 percent of its refinery staff were on strike and that its five refineries were storing their production while shipments were blocked by union picket lines.
"There is no risk of shortages," Total said.
Unions have vowed to halt the reforms, which are a central part of President Emmanuel Macron's agenda. Nationwide strikes and demonstrations have shut schools and closed transport services, while demonstrations have led to clashes with the police. Unions are planning a fourth day of nationwide demonstrations on Thursday.
(Reporting by Sudip Kar-Gupta, Benoit Van Overstraeten, Bate Felix and Geert De Clercq; Writing by Geert De Clercq; Editing by Giles Elgood and Alex Richardson)
By Sudip Kar-Gupta and Geert De Clercq
