A deal would create a combined company with a market valuation of about $47 billion and would be one of the biggest in recent years in the retail space.
Shares of Albertsons rose 7% on the report, while Kroger's stock slipped.
Merger talks between the two biggest US supermarket chains come at a time when retailers are seeing their margins squeezed from soaring costs and supply chain disruptions after a boom at the height of the pandemic
"It is hardly surprising," Neil Saunders, managing director of GlobalData Retail, said of the deal talks.
"Such consolidation would, in theory, improve the economics for both businesses giving them even more buying power than they currently enjoy."
Kroger, which also houses supermarket chains such as Fred Meyer, Ralphs and King Soopers, trails Walmart Inc, the top grocer in the country. Boise, Idaho-based Albertsons includes the Safeway banner.
A deal could be reached as soon as this week, Bloomberg reported, adding that no final decision has been taken and talks could still be delayed or falter.
Both companies did not immediately respond to Reuters' requests for comment on the report.
Ocado Shares Jump Over Merger Report
Elsewhere, shares in British online supermarket and technology group Ocado were up over 10% in late London trade on Thursday on hopes the company could benefit if US grocery giant Kroger Co merges with smaller rival Albertsons Companies Inc ACI.N.
Kroger is Ocado's biggest client.
Ocado struck a deal with Kroger in 2018 to help the US firm ratchet up its delivery business with the construction of robotically operated warehouses.
The initial deal saw Kroger identify at least 20 sites to build automated warehouses, or customer fulfilment centres (CFCs) as Ocado calls them, in the United States.
Ocado reported in May that six Kroger CFCs were live.
Ocado could not be immediately reached for comment.