CINCINNATI — The recent expansion of AmazonFresh in the Los Angeles market and other moves by grocers to offer on-line services are on the minds of executives at The Kroger Co. as the nation’s largest grocery store chain weighs the prospects of e-commerce.

In a June 20 conference call with analysts to discuss first-quarter financial results, Rodney McMullen, president and chief operating officer, said Kroger continues to work on getting the scale and profitability needed to make e-commerce a viable service for the company.

The company has been experimenting with on-line retailing and home delivery at its King Soopers stores in Denver for several years, and Mr. McMullen said the service is continuing to grow “at a modest rate.”

“We’re working on it, as much in terms of understanding the economics and trying to get a model that actually is profitable, so it’s one of those things where you should see us continue to work on it to improve it,” he said.

Mr. McMullen said the importance of e-commerce extends beyond just the products offered on-line, and to that end, Kroger is fully engaged.

“We’ve actually changed our team, our complete digital team in terms of bringing some high talent from the outside to help us accelerate our growth,” he said. “We feel very good about where we are headed, and as you know, from a digital standpoint, it’s broader than just what you sell on-line.

“We are also very aggressive on making sure we use our strength and insights we already have on our digital strategy, based on customer needs. We’re having huge growth in terms of the use of our web site, the use and download of our apps, and we’re also partnering with dunnhumby (a company that helps retailers and brands analyze data in order to improve customer experiences and build loyalty) to improve those from a relevancy standpoint.”

Asked whether Kroger envisions adopting a store pick-up type of program, Mr. McMullen said the company would consider “testing almost everything that you can imagine.”

“But I think at the end of the day, there will be something that will be a combination of a lot of different pieces together, and we actually think that’s one of the strengths that will play to us,” he said, noting that Kroger has many stores located close to customers’ houses.

In addition to Kroger and King Soopers, Kroger operates supermarkets under such banners as Dillons, Ralphs and Smith’s. In the first quarter ended May 25 Kroger had net income of $481 million, equal to 93c per share on the common stock, up from $439 million, or 78c per share, in the same period a year ago. Net sales in the first quarter were $30,043 million, up from $29,065 million.