It turns out those posts are fake.
The company is warning customers about the ad claiming to give away a year's worth of free groceries to a few lucky families.
It's posted by a fraudulent page called "Kroger-dot-com".
Kroger is warning the public not to click on the posts.
"This giveaway is fraudulent," the company wrote on Twitter. "Please do not click on it or provide any personal information. Our digital team is aware of this issue and is investigating."
Kroger, which owns Ralphs, Food 4 Less and other chains, is the country's biggest supermarket owner.
The official Facebook page for the company is just "@Kroger."
They're not the only store facing social media hoaxes.
Last month, Costco warned customers about a scam promising shoppers a $75 coupon.