Motorway blockades and picketing at refineries across the country have already hit British lorry drivers.
There are also fears industrial action could involve unions representing public transport staff and workers at airports and the Channel ports.
Dozens of petrol stations near Calais have already run dry, with the UK-based Freight Transport Association reporting that British truckers are being rationed.
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More strikes and protests are planned for tomorrow.
AA spokesman Paul Watters said: “It is well-known that labour disputes in France can affect peak season travel.
“Sometimes things get worse before they get better.”
Yesterday the hardline CGT labour union said that strikes had spread to all eight of France’s oil refineries.
It has also called weekly stoppages on state railways and an open-ended strike on the Paris underground and suburban train networks from June 2.
The unions are protesting over proposed labour reforms which could increase the working week and give companies more power to cut holiday allowances and maternity leave.