The results will be announced after voting ends on June 14, according to a statement by United Food and Commercial Workers Locals 8GS, 135, 324, 770, 1167, 1428 and 1442.
The contract expired Saturday, June 8.
The union's demands include equity, fair pay and safe stores.
"When we started contract negotiations with Food 4 Less/Foods Co., we made it clear that it (was) important to us that we reach a tentative agreement before contract expiration and we came to the table willing to put in the time and work to get that done," union officials said in a statement Saturday. "But instead of working with us towards a reasonable contract, our employer would rather play games with our livelihoods and offer proposals that grossly underestimate our value and their wealth.
"Everyone deserves a wage that reflects their work and no one deserves to be bullied at their job. By violating our rights, Food 4 Less and Kroger are making it harder to help our customers and keep our stores well- serviced, which is why we've been forced to take a strike authorization vote."
On May 22, the locals filed unfair labor practice charges against Food4less/Foods Co., alleging the company undermined negotiations and workers' rights to representation.
Union officials said the company has engaged in unlawful surveillance of workers, prohibited them from participating in union activity, blocked them from talking to union representatives and unilaterally changed the contract.
The union also claimed the company is attempting to strong-arm workers into accepting an offer that is less than what parent company Kroger provides other grocery workers in the area.
Food 4 Less, which has 15 stores in the Los Angeles area, is a subsidiary of The Kroger Company along with Ralphs.
"Over the last five years, Kroger and its family of stores have invested more than $2.4 billion in incremental wages," the company said in a statement earlier this year. "We look forward to negotiations with our goal to further reward our associates in ways that puts more money in their paychecks and invests in their future."