PARIS, Oct 10 (Reuters) - The strikes that have hit French oil refineries and storage sites will continue on Monday, CGT trade union representatives at TotalEnergies and ExxonMobil's Esso France told Reuters on Monday.

The strikes have added to pressure on President Emmanuel Macron's government, which is already facing discontent among the public over inflation and higher household bills.

"It will be continued across the board," said a CGT official at TotalEnergies, a day after the French company had offered to bring forward wage talks in an effort to end the labour dispute which has been dragging on for around two weeks.

The union said it needed to internally discuss TotalEnergies' offer to start wage talks earlier than initially planned.

Another CGT official said the walkouts would also continue at two Esso France refineries ahead of talks with management scheduled for later in the day.

"We still haven't got any confirmation that this meeting will be held," the union official said.

A spokeswoman for Esso France told Reuters the meeting will be held at the end of Monday morning and is set to include representatives from all four unions engaged at the company.

Almost a third of France's petrol stations had problems getting deliveries of at least one fuel product on Sunday, up from 21% the day, before according to the government.

(Reporting by Tassilo Hummel; Editing by Sudip Kar-Gupta)