SPRING HILL, Tenn.—When real-estate agent Don Nugent listed a three-bedroom, two-bath house here on Jo Ann Drive, offers came immediately, including a $208,000 one from a couple with a young child looking for their first home.

A competing bid was too attractive to pass up. American Homes 4 Rent, a public company that had been scooping up homes in the neighborhood, offered the same amount—but all cash, no inspection required.

Twelve...