As he prepares to take the reins as Macy’s chief executive on Sunday, Tony Spring has a tall order: He must contend with the existential crisis that mall-based department stores are facing to try to stay relevant in an increasingly e-commerce world.

But trying to infuse Macy’s with new ideas and win over the next generation of young shoppers shapes up as a long-term effort, and some investors have already lost their patience.

In December, an investor group submitted a bid that would take Macy’s private at a value of $5.8 billion. The investors, Arkhouse Management and Brigade Capital Management, say that unless the retailer begins sharing nonpublic information with them, they may take their offer to shareholders.

The firms’ bid puts even greater scrutiny on Mr. Spring as he takes over.

Mr. Spring, 58, has spent his career at Macy’s high-end store group, Bloomingdale’s. He started as an executive trainee in 1987 and rose through the ranks to become chief executive in 2014. During his tenure, he helped revive Bloomingdale’s cachet, bringing in hundreds of brands and emphasizing more appealing displays of merchandise and a creative marketing strategy. He turned the chain into a “scrappy incubator” for ideas that were eventually brought over to Macy’s stores.

Since March, Mr. Spring has been C.E.O.-elect of Macy’s Inc., which is the largest department store operator in the United States by revenue, once Bloomingdale’s and the cosmetics chain Bluemercury are factored in.

“Tony, I think, has done a better job, quite frankly, than the Macy’s organization has done in keeping their stores tuned into their customers and making the presentation sparkle, be enticing and making people want to walk to the store to get ideas,” said Allen Questrom, the former chief executive of Federated Department Stores, as Macy’s parent company was then known.