Circle K owner Alimentation Couche-Tard Inc. is nearing a deal to buy CST Brands Inc. in a big move to consolidate the convenience-store industry.

Couche-Tard is the likely winner of an auction of CST, according to people familiar with the matter. It isn’t clear what the Canadian company would pay, but CST has a market value of $3.4 billion, meaning a deal is likely to be valued around that level or higher.

A...