Negotiations between the union and the owners of Ralphs, Vons and Albertsons resumed on Monday with the help of a federal mediator.
Earlier this month, more than 90 percent of the union's 62,000 grocery workers represented by the United Food and Commercial Workers union voted to authorize a strike.
Some of the stores have been accepting applications for fill-in workers in case the grocery workers do walk off the job.
The workers' contract expired six months ago. Health care is the main sticking point.
Under the most recent industry offer, workers would pay about $36 per month for individual health insurance, or $92 per month for family coverage. But the union says the insurance fund would not be financially viable and wants the supermarkets to contribute more to prevent it from running out of money within 16 months.
No tentative agreement on wages has been reached.
City News Service contributed to this report.